Department of Mathematicshttp://hdl.handle.net/1956/1065
Tue, 26 Sep 2017 21:31:11 GMT2017-09-26T21:31:11Z«Hæh? Er det der en modell liksom?» Ei studie av R2 elevar si oppfatning av læringsutbyttet ved eit didaktisk kontraktbrot.http://hdl.handle.net/1956/16432
«Hæh? Er det der en modell liksom?» Ei studie av R2 elevar si oppfatning av læringsutbyttet ved eit didaktisk kontraktbrot.
Kjepso, Peter
Master thesis
Matematikkundervisning i Noreg følgjer stort sett same didaktiske kontrakt. Korleis opplever elevar med full fordjuping i faget eit brot på denne kontrakten? Studia baserer seg på to teoretiske fundament: Det teoretiske grunnlaget for modellering som metode samt tankane til Brousseau rundt den didaktiske situasjonen.
Wed, 26 Jul 2017 00:00:00 GMThttp://hdl.handle.net/1956/164322017-07-26T00:00:00ZAutomorphism groups of pseudo H-type Lie algebrashttp://hdl.handle.net/1956/16377
Automorphism groups of pseudo H-type Lie algebras
Azzolini, Francesca
Master thesis
Wed, 26 Jul 2017 00:00:00 GMThttp://hdl.handle.net/1956/163772017-07-26T00:00:00ZGeometric and Spectral Properties of Hypoelliptic Operatorshttp://hdl.handle.net/1956/16376
Geometric and Spectral Properties of Hypoelliptic Operators
Eik, Stine Marie
Master thesis
Wed, 26 Jul 2017 00:00:00 GMThttp://hdl.handle.net/1956/163762017-07-26T00:00:00ZOn extremal function in one class of mapping with finite distortionhttp://hdl.handle.net/1956/16375
On extremal function in one class of mapping with finite distortion
Ingale, Bhagyashri Nilesh
Master thesis
Wed, 26 Jul 2017 00:00:00 GMThttp://hdl.handle.net/1956/163752017-07-26T00:00:00ZElevers begrepsforståelse i modelleringskonteksten - En kvalitativ studie av tre elevers begrepsforståelse i arbeidet med autentiske modelleringsoppgaverhttp://hdl.handle.net/1956/16374
Elevers begrepsforståelse i modelleringskonteksten - En kvalitativ studie av tre elevers begrepsforståelse i arbeidet med autentiske modelleringsoppgaver
Ommedal, Eili Seljeset
Master thesis
Målet med studien var å få bedre forståelse for hvordan elevers arbeid med en autentisk modelleringsoppgave kunne påvirke begrepsforståelsen. Jeg har i denne studien valgt å fokusere på begrepet den andrederiverte. I studien ble det benyttet kvalitative forskningsmetoder i form av observasjon og intervju. Som en del av intervjuet ble det også gitt oppgaver om den andrederiverte. Dette for å få et bedre innblikk i elevenes begrepsforståelse. Utvalget besto av tre elever som har programfaget R1 på videregående skole. Datamaterialet er samlet inn i løpet av tre uker, der det ble gjennomført fem økter med modelleringsarbeid samt tre intervjuer. I løpet av observasjonen og i intervjuene ble det fokusert på hvordan begrepsforståelsen ble påvirket. Det ble valgt ut samtalesekvenser fra datamaterialet til analysene, og sekvensene ble analysert ved hjelp av et analyseverktøy som består av teoretiske begreper om begrepsforståelse. Studiens funn antyder at det er fire sentrale faktorer ved modelleringsarbeidet som påvirker begrepsforståelsen; Samtalen, det autentiske aspektet, arbeid uten en kjent framgangsmåte eller fasit og bruk av digital graftegner. Når elevene deltar i faglige samtaler, kan dette føre til at de gjør viktige refleksjoner i tilknytning til de begrepene som omhandles. Det autentiske med modellering bidrar til en konkretisering, slik at de matematiske begrepene i større grad kan gi mening fordi de knyttes direkte til elevenes erfaringer. Den tredje faktoren, arbeid uten en kjent framgangsmåte eller fasit, gjør at de i større grad må tenke over hvordan de skal løse problemet og derfor kreves det at en må forstå hva begrepene betyr. For den siste faktoren, bruk av digital graftegner, gir denne en spesiell mulighet til visualisering av begrepene. Funn viser at modelleringen ser ut til å påvirke de tre elevenes begrepsforståelse på ulike måter. I denne sammenhengen indikerer resultatene at modelleringsarbeidets påvirkning på begrepsforståelsen, er tett knyttet til elevenes faglige og sosiale trygghet. Når elevene er trygge kan de tørre å ta steget ut i faglige diskusjoner, noe som kan være utviklende for begrepsforståelsen. På bakgrunn av resultatene, kan det argumenteres for at det i forkant av et modelleringsprosjekt bør arbeides med å øke tryggheten til elevene innenfor de fire ovenfornevnte faktorene. I selve modelleringsprosjektet bør også læreren stille elevene utfordrende spørsmål, noe som kan bidra til at elevene gjør viktige refleksjoner omkring begrepene. Disse momentene ser ut til å være sentrale for å hjelpe elevene med å utvikle en bedre forståelse av begrepene som inngår i modelleringsarbeidet.
Wed, 26 Jul 2017 00:00:00 GMThttp://hdl.handle.net/1956/163742017-07-26T00:00:00ZVisualisering og konkretisering av omdreiningslegemer ved bruk av Geogebra 3D og 3D-printerhttp://hdl.handle.net/1956/16373
Visualisering og konkretisering av omdreiningslegemer ved bruk av Geogebra 3D og 3D-printer
Bakken, Vibeke Synøve
Master thesis
Digital programvare og digitale verktøy er i stadig utvikling, og matematiske programmer, som for eksempel Geogebra, gir oss mulighet til å visualisere matematikk på en enklere og raskere måte enn tidligere, både gjennom todimensjonale og tredimensjonale representasjoner. De senere år har også 3D-printere blitt tatt i bruk i flere videregående skoler, og dette gir nye muligheter for konkretisering av matematikk. Visualisering og konkretisering er ikke ukjente virkemidler i matematikkfagene, men mitt ønske var å undersøke læringseffekten som eventuelt oppnås ved å bruke digitale hjelpemidler til visualisering og konkretisering i programfagene i matematikk. Jeg valgte et emne fra læreplanen i matematikk R2 som egner seg godt for visualisering og konkretisering, omdreiningslegemer, og jeg arbeidet ut fra følgende problemstilling: Hvordan opplever elever at digital 3D-visualisering og konkretisering med 3D-printer bidrar til utvidet forståelse av omdreiningslegemer? For å prøve å finne svar på denne problemstillingen, gjennomførte jeg en fagdag i matematikk R2 der visualisering og konkretisering med digitale hjelpemidler hadde hovedfokus. Fagdagen inneholdt en kort teoretisk gjennomgang, der visualisering, animasjoner og konkreter ble benyttet, og deretter arbeidet elevene med oppgaver. Første oppgave tok utgangspunkt i et konkret plastglass som elevene skulle modellere og visualisere ved hjelp av Geogebra. Deretter skulle de beregne volum av sin «modell» ved hjelp av CAS, og ett eksemplar skulle så konkretiseres ved hjelp av 3D-printer. Til slutt fikk de en kreativ oppgave som gikk ut på at hver elev skulle modellere et eget drikkebeger ut fra ett eller flere funksjonsuttrykk og omdreining av disse. Drikkebegrene skulle visualiseres ved hjelp av Geogebra 3D for så å bli konkretisert ved hjelp av 3D-printer. Resultatene av denne studien tyder på at elevene opplever at det å bruke digitale hjelpemidler for å visualisere og konkretisere, gjør at de får økt forståelse for hva omdreiningslegemer er. Gjennom å eksperimentere med enkle og sammensatte funksjoner ble de tryggere i sin bruk av Geogebra, og de lærte en metode for å visualisere omdreiningsfigurer tredimensjonalt som de så nytte av i flere sammenhenger. Elevene ga også tilbakemeldinger om at de har hatt en ny og positiv opplevelse i forhold til matematikkfaget; matematikk var ikke lenger bare «noe på papiret», de hadde erfart at matematikk kan bli til noe konkret.
Wed, 26 Jul 2017 00:00:00 GMThttp://hdl.handle.net/1956/163732017-07-26T00:00:00ZOmvendt undervisning i matematikk. En kvalitativ studie av hvordan elevene benytter seg av video komponenten av omvendt undervisning. Hva sier en slik bruk av video om fordeler og ulemper med omvendt undervisning.http://hdl.handle.net/1956/16372
Omvendt undervisning i matematikk. En kvalitativ studie av hvordan elevene benytter seg av video komponenten av omvendt undervisning. Hva sier en slik bruk av video om fordeler og ulemper med omvendt undervisning.
Indrebø, Lena
Master thesis
Da jeg bestemte meg for å skrive masteroppgave i matematikkdidaktikk tenkte jeg da at omvendt
undervisning ville være et spennende tema. Problemstillingen min ble som følger:
Hvordan benytter elevene seg av video i forbindelse med omvendt undervisning i matematikk?
Hva sier en slik bruk av video om fordeler og ulemper med omvendt undervisning i forhold til
tradisjonell undervisning?
Gjennom den første delen av problemstillingen så jeg på hvilke læringsstrategier elevene benyttet
seg av når de så en video og gjorde en tilhørende oppgave. Jeg så også på hvilken forståelse de hadde
fokus på. Gjennom den andre delen av problemstillingen diskuterte jeg om metoden omvendt
undervisning kunne være årsak til noen av strategiene som elevene benyttet. Jeg diskutere hva en
slik bruk av video viste om fordeler og ulemper med omvendt undervisning i forhold til tradisjonell
undervisning.
For å svare på problemstillingen benyttet jeg meg av observasjon og kvalitativt intervju av 4 elever
fra en R2 klasse.
Jeg observerte elevene se en video og løse en oppgave. Jeg intervjuet deretter elevene om hvordan
de hadde tenkt gjennom oppgaven. På denne måten fikk jeg innblikk i hvilke læringsstrategier de
benyttet og hvilken type forståelse de satt igjen med etter videoen. Jeg stilte også elevene noen
generelle intervjuspørsmål for å få mer innblikk i hvordan de generelt benyttet seg av video og
hvordan de opplevde omvendt undervisning i forhold til tradisjonell undervisning.
Undersøkelsen viste både fordeler og ulemper med omvendt undervisning med tanke på hvordan
elevene benyttet seg av video. Omvendt undervisning ga hver enkelt elev mulighet til å styre tempo
på gjennomgangen av et nytt tema. På den måten kunne elevene ta seg den tiden de trengte til å
benytte gode læringsstrategier og få en god forståelse. På en annen side var det to elever som ga
meg litt inntrykk av å bare ha fokus på å lære fremgangsmåtene som presenteres i videoen. Det
kunne komme av oppbygningen av videoen. Det var ikke nødvendigvis konseptet omvendt
undervisning som førte til det.
Det kom fram at omvendt undervisning kanskje kan være et bra alternativ i forhold til tradisjonell
undervisning. Tradisjonell undervisning bærer ofte preg av mye direkte instruksjon og lite
elevaktivitet. Omvendt undervisning vil også innebære dirkete instruksjon gjennom videoen. Når
elevene er forberedt med en video er det kanskje lettere for dem å engasjere seg i elevaktiviteter på
skolen hvor de må reflektere over kunnskapen de fikk i video.
Wed, 26 Jul 2017 00:00:00 GMThttp://hdl.handle.net/1956/163722017-07-26T00:00:00ZAnalyse av undervisningsopplegg med undersøkende matematikkhttp://hdl.handle.net/1956/16371
Analyse av undervisningsopplegg med undersøkende matematikk
Christensen, Tore Morland
Master thesis
I denne masteroppgaven ser jeg på hvilken effekt undersøkende matematikk kan ha på
elevers holdninger til faget med hovedvekt på motivasjon. Prosjektene har blitt
gjennomført med ungdomsskoleelever ved Stiftelsen Montessoriskolen i Bergen.
Wed, 26 Jul 2017 00:00:00 GMThttp://hdl.handle.net/1956/163712017-07-26T00:00:00ZLog Hochschild Homology of Categorical Products and Coproductshttp://hdl.handle.net/1956/16370
Log Hochschild Homology of Categorical Products and Coproducts
Vågset, Erlend Raa
Master thesis
As the front page implies, this is a thesis about log Hochschild homology. Our primary goal will
be to understand how the log Hochschild homology groups acts on products and coproducts in
the category of commutative pre-log algebras. We will prove that the log Hochschild homology
commutes with products, and we shall formulate and prove a K¨unneth Theorem for log Hochschild
homology. These results allows for the computation of the log Hochschild homology of complicated
pre-log algebras that have been constructed from components we already understand by the process
of taking products and coproducts.
Wed, 26 Jul 2017 00:00:00 GMThttp://hdl.handle.net/1956/163702017-07-26T00:00:00ZSkriving i matematikk. Hva skriver elevene? En studie av elevenes fagspråk i matematikk 2P med eksempler fra emnet funksjoner.http://hdl.handle.net/1956/16369
Skriving i matematikk. Hva skriver elevene? En studie av elevenes fagspråk i matematikk 2P med eksempler fra emnet funksjoner.
Lehtinen, Hilde
Master thesis
I masteroppgaven ”Skriving i matematikk. Hva skriver elevene? En studie av fagspråk i
matematikk 2P med eksempler fra emnet funksjoner” fokuseres det på analyse av
elevbesvarelser. Formålet med denne studien er å få økt innsikt i hva elevene skriver i faget
matematikk 2P og hvordan elevene kan synliggjøre sin kunnskap gjennom skriving i faget.
Det anvendes kvalitativ tilnærming med innsamling av elevbesvarelser på utvalgte oppgaver
som instrument. Utvalget av elevbesvarelser er gjort blant egne elever og det er også
utarbeidet anvendbare kategorier for å kategorisere elevbesvarelsene. Undersøkelsene i
studien er gjort på en allmenfaglig skole i matematikktimene. Elevenes mestringsnivå har
dekket hele karakterskalaen med 1 som laveste karakter og 6 som høyeste karakter. I
gruppene jeg har jobbet med har det vært stor spredning i karakter og min erfaring tilsier at
gruppene mine har vært representative for denne typen grupper. Anonymitet er sikret ved at
studien er meldt til Norsk samfunnsfaglig datatjeneste (NSD) og godkjent som meldepliktig
siden det også samles inn digitale elevbesvarelser. Resultatene er presentert med
tekstanalyse og noen få søylediagram.
I det teorietiske grunnlaget for studien er det fokusert på skriving som grunnleggende
ferdighet, matematikksrpåkets egenart og fagskriving i matematikk. Tanken bak studien er at
skriving er et viktig verktøy for å lære, systematisere og å synliggjøre kunnskap i arbeidet med
matematikkfaget. Resultatene av studien viser at det å gå i mellom ulike representasjoner som
tekst, graf, tabell og funksjonsuttrykk nok medfører en del uklarheter og usikkerheter for
elevene. Resultatene viser også at grafen er nok den representasjonen som gir elevene mest
informasjon og at det gjerne trengs litt jobb for at de andre representasjonene skal få samme
verdi for elevene.
Wed, 26 Jul 2017 00:00:00 GMThttp://hdl.handle.net/1956/163692017-07-26T00:00:00ZExistence and Uniqueness of Non-linear, Possibly Degenerate Parabolic PDEs, with Applications to Flow in Porous Mediahttp://hdl.handle.net/1956/16368
Existence and Uniqueness of Non-linear, Possibly Degenerate Parabolic PDEs, with Applications to Flow in Porous Media
Westrheim, Anders
Master thesis
In the branch of mathematical analysis known as functional analysis, one mainly studies functions defined on vector spaces. For partial differential equations (PDEs), this analysis has proven to be a mighty resource of understanding and modelling the behavior of the equations. Throughout this thesis, the work will focus of theory of function spaces and existence and uniqueness theorems for variational formulations in normed vector spaces. We will recast PDEs as variational problems with operators acting on normed spaces, and further seek to prove the existence and uniqueness of a solution by assigning certain properties to the operator. The outline of this thesis is as follows: In Chapter 1, we summarize the Basic Notions of Functional Analysis relevant for the later work in the thesis. We define operators, discuss monotonicity, present the theory of Sobolev spaces, and illustrate the finite element method, giving short hints to the future relevancy of the described properties. Linear Problems have been extensively studied in the past. In Chapter 2, we present three important theorems illustrating the conditions for existence and uniqueness of solutions for variational formulations of the type: (i) Galerkin formulations in Hilbert spaces: The Lax-Milgram Theorem, (ii) Petrov-Galerkin formulations in Hilbert spaces: The Babuška-Lax-Milgram Theorem, (iii) Petrov-Galerkin formulations in Banach spaces: The Banach-Nečas-Babuška Theorem, and give their proofs. Chapter 3 is dedicated to the study of Non-linear Problems. We seek to extend the ideas of the previous chapter to variational formulations containing a non-linearity b(·) depending on the solution we seek. This has a major application in the analysis of non-linear PDEs, which in general may not possess analytical solutions. To attack these types of problems, we define a weak formulation of the main problem, and discretize the domain of where a solution is sought. Next, existence and uniqueness is established through fixed point theorems, which will be given with proof. We will focus our study on two central problems: The Richards equation (a non-linear, possibly degenerate parabolic PDE) and a transport equation modelling reactive flow in porous media (two coupled PDEs). For the fully discrete (non-linear) formulation of Richards equation we show results for (i) a Lipschitz continuous non-linearity. Here we consider three cases: 3(a) First, a linearization scheme is proposed. We prove existence and uniqueness by using the Lax-Milgram Theorem in combination with the Banach Fixed Point Theorem. (b) Second, we make the assumption that the non-linearity is strongly monotone. Here, existence is proven by the Brouwer Fixed Point Theorem (c) Third, we let the non-linearity be monotone and add a regularization term to the fully discrete formulation. Here, we prove existence as in the previous step, and lastly show convergence of the regularized scheme to the fully discrete scheme. (ii) a Hölder continuous non-linearity. We give two results: (a) First, we prove existence for a monotone and bounded non-linearity. (b) Second, we state the result of existence for a strongly monotone non-linearity by the Brouwer Fixed Point Theorem. In the applications of Brouwer Fixed Point Theorem, the uniqueness of the problem is proved by assuming there exists two solutions and obtaining a contradiction through inequalities by showing estimates that can not be true. Lastly, in Chapter 4, a mathematical model of Two-phase Flow in porous media is studied. We discuss the case of a Lipschitz continuous saturation, and show for the first time a proof of existence and uniqueness of a solution for the fully discrete (non-linear) scheme, assuming the saturation to be Hölder continuous and strongly monotonically increasing. This is done by creating a regularization of the fully discrete scheme, further proving existence with the Brouwer Fixed Point Theorem, and finally showing convergence with the help of an a priori estimate.
Wed, 26 Jul 2017 00:00:00 GMThttp://hdl.handle.net/1956/163682017-07-26T00:00:00ZDerivation and numerical solution of fully nonlinear and fully dispersive water wave model equationshttp://hdl.handle.net/1956/16254
Derivation and numerical solution of fully nonlinear and fully dispersive water wave model equations
Moldabayev, Daulet
Doctoral thesis
<p>The water wave theory is a classical part of Fluid Mechanics. It has a long scientific history with a great number of mathematical results [10, 32]. The first studies in this field were done by Stokes in 1847 [34]. He developed some approximations to periodic waves and proposed conjectures about their behavior on deep water. Today these waves are known as Stokes waves.</p>
<p>The problem of water waves concerns the two-dimensional flow of an inviscid, incompressible fluid, bounded above by a free surface and below by a rigid horizontal bottom. In this situation, the flow is described by Euler equations with appropriate boundary conditions [13]. By solving these equations, one obtains a complete understanding of the flow dynamics. However, for some applications the dynamics of the free surface is of particular interest. Nonlinear dispersive wave equations, such as the Korteweg&ndash;de Vries equation [22], allow to approximate the description of the free surface evolution without having to provide a complete solution of the fluid flow below the surface. Different questions related to these equations are actively researched. The existence of traveling and solitary waves solutions [15], well posedness of these equations [21, 24, 25] are two examples.</p>
<p>In this work, nonlinear dispersive water wave equations are analyzed from a number of points: their accuracy in approximating the solutions of Euler equations, derivation from Hamiltonian formulation of the water wave problem, numerical solution of these equations, stability of their solutions and investigation of their bifurcation curves.</p>
<p>The Part I of this thesis consists of three chapters. Chapter 1 is devoted to mathematical formulation of the surface water-wave problem. We revise the Eulerian formulation of the problem, the Linear water-wave theory and the boundary conditions applied. The second chapter describes a method for deriving nonlinear dispersive water-wave equations. The main focus here is on the Hamiltonian formulation of the problem and different scaling regimes. A numerical method for solving the water-wave equations is also presented in Chapter 2. Chapter 3 gives a brief summary of the research results obtained in the course of the doctoral studies. Part II contains the research papers, published and submitted for publication, which were written for this PhD project.</p>
Fri, 09 Jun 2017 00:00:00 GMThttp://hdl.handle.net/1956/162542017-06-09T00:00:00ZA numerical study of nonlinear dispersive wave models with SpecTraVVavehttp://hdl.handle.net/1956/16253
A numerical study of nonlinear dispersive wave models with SpecTraVVave
Kalisch, Henrik; Moldabayev, Daulet; Verdier, Olivier
Journal article
In nonlinear dispersive evolution equations, the competing effects of nonlinearity and dispersion make a number of interesting phenomena possible. In the current work, the focus is on the numerical approximation of traveling-wave solutions of such equations. We describe our efforts to write a dedicated Python code which is able to compute traveling-wave solutions of nonlinear dispersive equations in a very general form.
The SpecTraVVave code uses a continuation method coupled with a spectral projection to compute approximations of steady symmetric solutions of this equation. The code is used in a number of situations to gain an understanding of traveling-wave solutions. The first case is the Whitham equation, where numerical evidence points to the conclusion that the main bifurcation branch features three distinct points of interest, namely a turning point, a point of stability inversion, and a terminal point which corresponds to a cusped wave.
The second case is the so-called modified Benjamin-Ono equation where the interaction of two solitary waves is investigated. It is found that two solitary waves may interact in such a way that the smaller wave is annihilated. The third case concerns the Benjamin equation which features two competing dispersive operators. In this case, it is found that bifurcation curves of periodic traveling-wave solutions may cross and connect high up on the branch in the nonlinear regime.
Thu, 02 Mar 2017 00:00:00 GMThttp://hdl.handle.net/1956/162532017-03-02T00:00:00ZVisuelle løsningsstrategier til sannsynlighetsoppgaver - En studie av fem videregående elevers bruk av ulike visuelle løsningsstrategier i sannsynlighetsregninghttp://hdl.handle.net/1956/16152
Visuelle løsningsstrategier til sannsynlighetsoppgaver - En studie av fem videregående elevers bruk av ulike visuelle løsningsstrategier i sannsynlighetsregning
Meidell, Camilla
Master thesis
Denne masteroppgaven handler om elever på videregående sin bruk av visuelle løsningsstrategier i møte med sannsynlighetsoppgaver. Formålet var blant annet å undersøke hvilke ulike visuelle løsningsstrategier elever benytter seg av i møte med slike oppgaver. De andre aspektene jeg ønsket å studere var hvorfor elevene valgte strategiene og hvordan bruken av visuelle løsningsstrategier kan sees i sammenheng med misoppfatninger innenfor sannsynlighet. I studien er det blitt brukt en blandet metode. Først ble det gjennomført en kvantitativ spørreundersøkelse hvor 38 elever deltok. Resultatene fra denne undersøkelsen ble så brukt til å velge ut fem elever til fokusgruppeintervju. Elevene var hentet fra 1P og R2 og hadde således ulik matematisk bakgrunn. Spørsmålene som ble presentert for elevene er hentet fra tidligere undersøkelser, dermed kan en anta at spørsmålene har gjennomgått kvalitetssikring. Resultatet til undersøkelsen viser at elevene bruker noen av de visuelle løsningsstrategiene som omtales i litteraturen. Av de visuelle løsningsstrategiene var det utfallslisting som ble brukt mest. Det kommer frem i intervju med elevene at noen bruker visuelle løsningsstrategier fordi de mener at dette forenkler løsningsprosessen. Andre bruker visuelle løsningsstrategier når de er usikker på oppgaven. Resultatene tyder på at det noen ganger kan hjelpe elevene å unngå de typiske feilene i sannsynlighetsregning om de benytter seg av visuelle løsningsstrategier. Det ser for eksempel ut som om visuelle løsningsstrategier, kan klargjøre utfallsrommet for elevene. Det kommer imidlertid også frem at det er noen utfordringer knyttet til bruken av de visuelle løsningsstrategiene som kan resultere i at elevene til tross for bruk av visuelle løsningsstrategier ikke klarer å se for seg hele utfallsrommet. Disse utfordringene ser ut til å finnes i sammenheng med misoppfatninger så vel som i forbindelse med manglende kunnskaper om strategien i seg selv.
Wed, 26 Jul 2017 00:00:00 GMThttp://hdl.handle.net/1956/161522017-07-26T00:00:00ZA convergent mass conservative numerical scheme based on mixed finite elements for two-phase flow in porous mediahttp://hdl.handle.net/1956/15970
A convergent mass conservative numerical scheme based on mixed finite elements for two-phase flow in porous media
Radu, Florin Adrian; Kumar, Kundan; Nordbotten, Jan Martin; Pop, Sorin Iuliu
Research report
In this work we present a mass conservative numerical scheme for two-phase flow in porous media. The
model for flow consists on two fully coupled, non-linear equations: a degenerate parabolic equation and an elliptic
equation. The proposed numerical scheme is based on backward Euler for the temporal discretization and mixed
finite element method (MFEM) for the discretization in space. Continuous, semi-discrete (continuous in space)
and fully discrete variational formulations are set up and the existence and uniqueness of solutions is discussed.
Error estimates are presented to prove the convergence of the scheme. The non-linear systems within each time
step are solved by a robust linearization method. This iterative method does not involve any regularization step.
The convergence of the linearization scheme is rigorously proved under the assumption of a Lipschitz continuous
saturation. Numerical results are presented to sustain the theoretical findings.
Sun, 01 Jan 2017 00:00:00 GMThttp://hdl.handle.net/1956/159702017-01-01T00:00:00ZOn the Relation between Surface Profiles and Internal Flow Properties in Long-Wave Modelshttp://hdl.handle.net/1956/15946
On the Relation between Surface Profiles and Internal Flow Properties in Long-Wave Models
Senthilkumar, Amutha
Doctoral thesis
<p>In this work, we investigate the internal velocity field in a number of Boussinesq models in non-uniform situations. A coupled BBM-BBM type system of equations is derived in the assumption of water wave propagating over an uneven bottom. The focus is on formulating mass, momentum and energy densities and fluxes associated with the BBM-BBM system over an uneven bottom. These densities and the associated fluxes arise from establishing mechanical balance equations of the same asymptotic order as the evolution equations.</p>
<p>The BBM-BBM type system derived here is solved numerically by applying a Fourier collocation method coupled with a four stage Runge-Kutta time integration scheme. We look at the propagation of waves over a slope, and how the reconstruction of the flow under the surface is connected with shoaling and wave breaking. The mass conservation equations are used to quantify the role of reflection in the shoaling of solitary waves. Moreover, the principle of conservation of energy is used to develop an equation relating the waveheight and undisturbed depth to the initial undisturbed depth and the incident waveheight. Boussinesq&rsquo;s shoaling law is approximately recovered for waves of very small waveheight. Shoaling and breaking results for the different Boussinesq systems are plotted.</p>
<p>Internal properties of the flow are also in focus in the case of a background shear flow. The Boussinesq -type equations for water waves with constant vorticity are derived in the Boussinesq regime. We reduced the Boussinesq -type equations to the Korteweg-de Vries (KdV) equation in the unidirectional case. We found the approximate velocity field associated with exact solutions of KdV equation including shear flow. The influence of the shear flow on particle trajectories and breaking of surface waves are studied using the approximate velocity field.</p>
Fri, 10 Mar 2017 00:00:00 GMThttp://hdl.handle.net/1956/159462017-03-10T00:00:00ZModelling Geomechanical Impact of CO2 Injection Using Precomputed Response Functionshttp://hdl.handle.net/1956/15945
Modelling Geomechanical Impact of CO2 Injection Using Precomputed Response Functions
Andersen, Odd; Nilsen, Halvor Møll; Gasda, Sarah
Conference object
When injecting CO2 or other fluids into a geological formation, pressure plays an important role both as a driver of flow and as a risk factor for mechanical integrity. The full effect of geomechanics on aquifer flow can only be captured using a coupled flow-geomechanics model. In order to solve this computationally expensive system, various strategies have been put forward over the years, with some of the best current methods based on sequential splitting. In this present work, we seek to approximate the full geomechanics effect on flow without the need of coupling with a geomechanics solver during simulation. We do this by means of precomputed pressure response functions. At grid model generation time, a geomechanics solver is used to compute the mechanical response of the aquifer for a set of pressure fields. The relevant information from these responses is then stored in a compact form and embedded with the grid model. We test the accuracy and computational performance of our approach on a simple 2D model and a more complex 3D model, and compare the results with those produced by a fully coupled approach as well as from as simple decoupled method based on Geertsma's uniaxial expansion coefficient.
Presented at: ECMOR XV - 15th European conference on the mathematics of oil recovery, Amsterdam, Netherlands, 29 August-1 September 2016
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/1956/159452016-01-01T00:00:00ZOn the influence of wave reflection on shoaling and breaking solitary Waveshttp://hdl.handle.net/1956/15944
On the influence of wave reflection on shoaling and breaking solitary Waves
Senthilkumar, Amutha
Journal article
A coupled BBM system of equations is studied in the situation of water waves propagating over a decreasing fluid depth. A conservation equation for mass and also a wave breaking criterion, both valid in the Boussinesq approximation, are found. A Fourier collocation method coupled with a 4-stage Runge–Kutta time integration scheme is employed to approximate solutions of the BBM system. The mass conservation equation is used to quantify the role of reflection in the shoaling of solitary waves on a sloping bottom. Shoaling results based on an adiabatic approximation are analysed. Wave shoaling and the criterion of the breaking of solitary waves on a sloping bottom are studied. To validate the numerical model the simulation results are compared with reference results and a good agreement between them can be observed. Shoaling of solitary waves is calculated for two different types of mild slope model systems. Comparison with reference solutions shows that both of these models work well in their respective regimes of applicability.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/1956/159442016-01-01T00:00:00ZDerivation of Boussinesq's shoaling law using a coupled BBM systemhttp://hdl.handle.net/1956/15943
Derivation of Boussinesq's shoaling law using a coupled BBM system
Kalisch, Henrik; Senthilkumar, Amutha
Journal article
This paper is focused on finding rules for waveheight change in a solitary wave as it runs up a slowly increasing bottom. A coupled BBM system is used to describe the solitary waves. Expressions for energy density and energy flux associated with the BBM system are derived, and the principle of energy conservation is used to develop an equation relating the waveheight and undisturbed depth to the initial undisturbed depth and the incident waveheight. In the limit of zero waveheight, Boussinesq's shoaling law is recovered.
Thu, 14 Mar 2013 00:00:00 GMThttp://hdl.handle.net/1956/159432013-03-14T00:00:00ZSurvey strategies to quantify and optimize detecting probability of a CO2 seep in a varying marine environmenthttp://hdl.handle.net/1956/15921
Survey strategies to quantify and optimize detecting probability of a CO2 seep in a varying marine environment
Hvidevold, Hilde Kristine; Alendal, Guttorm; Johannessen, Truls; Ali, Alfatih Omer Mohammed Ahmed
Journal article
Designing a marine monitoring program that detects CO2 leaks from subsea geological storage projects is challenging. The high variability of the environment may camouflage the anticipated anisotropic signal from a leak and there are a number of leak scenarios. Marine operations are also costly constraining the availability of measurements. A method based on heterogeneous leak scenarios and anisotropic predictions of chemical footprint under varying current conditions is presented. Through a cost function optimal placement of sensors can be given both for fixed installations and series of measurements during surveys. Ten fixed installations with an optimal layout is better than twenty placed successively at the locations with highest leakage probability. Hence, optimal localizations of installations offers cost reduction without compromising precision of a monitoring program, e.g. quantifying and reduce probabilities of false alarm under control. An optimal cruise plan for surveys, minimizing transit time and operational costs, can be achieved.
Thu, 01 Sep 2016 00:00:00 GMThttp://hdl.handle.net/1956/159212016-09-01T00:00:00ZComparing Maximum Likelihood and Generalized Method of Moments in Short Term Interest Rate Modelshttp://hdl.handle.net/1956/15812
Comparing Maximum Likelihood and Generalized Method of Moments in Short Term Interest Rate Models
Hanevik, Severin
Master thesis
In this thesis we will look at some different continuous models for predicting the short term interest rate, and focus on the method of parameter estimation in such models. A particular focus will be placed on the method of Maximum Likelihood Estimation (MLE), which has not been the most common method of estimation in this context. Furthermore, we will compare MLE to the Generalized Method of Moments (GMM), and both methods will be used in simulation experiments and on different sets of real data. Our starting point for investigation will be the model used in Chan et al.(1992), a model in which several of the most popular short rate models can be nested. We will use a discrete-valued approximation of this model to facilitate the estimation of parameters.
In conclusion, this thesis argues that the MLE method of parameter estimation in short term interest rate models deserves more attention than it is currently given. In simulation experiments, the MLE method produced more accurate and precise estimates than the GMM method. Specifically, the bias in estimating the mean-reversion parameter is smaller using the MLE method. These results are particularly interesting as the GMM method is currently the common approach to parameter estimation in short term interest rate models. However, as will be shown in this thesis, it may very well be that the MLE method constitutes a better approach than GMM for such estimations. The results in this thesis may therefore contribute to the theoretical perspective as well as the real-world applications of methods of parameter estimation in predicting the short term interest rate.
Mon, 01 Feb 2016 00:00:00 GMThttp://hdl.handle.net/1956/158122016-02-01T00:00:00ZProblemløsning i norske og russiske matematikklærebøker for videregående skole. En sammenlignende studie av eksempler i norske og russiske lærebøkerhttp://hdl.handle.net/1956/15811
Problemløsning i norske og russiske matematikklærebøker for videregående skole. En sammenlignende studie av eksempler i norske og russiske lærebøker
Aaseth, Natalia
Master thesis
Denne masteroppgaven sammenligner lærebøkene
fra to land, Norge og Russland, i emneområder
Algebra og Funksjoner, og fokuserer på
problemløsningsmetoder i eksemplene i de norske
og de russiske lærebøkene for videregående
skole. Russiske elever presterer bedre enn
norske elever i forskjellige internasjonale
undersøkelser, som TIMSS og PISA, og det var
interessant å finne ut hva det skyldes. Målet
med denne studien har vært å undersøke hva som
skiller læreplanene og lærebøkene fra de to
landene, når det gjelder problemløsning og
problemløsningsmetoder.
Problemstillingen i denne masteroppgaven har
vært følgende: Hva er forskjeller og likheter i
presentasjonen av problemløsningsmetoder i
eksemplene i norske og russiske
matematikklærebøker?
For å svare på problemstillingen brukte jeg
kombinasjon av kvalitative og kvantitative
metoder med dokumentanalyse som
datainnsamlingsmetode. Analysen begrenses til
eksemplene i de norske lærebøkene for
matematikkursene 1T og R1, og til eksemplene i
de russiske lærebøkene for 10. og 11. trinn.
Først analyserte jeg den norske læreplanen og
den russiske læreplanen for å sammenligne deres
syn på problemløsning. Analysen av lærebøker
starter med en liten hjelpeundersøkelse for å
vise at de undersøkte lærebøkene er
sammenlignbare, jeg brukte rammeverket til Hong
og Choi og tilpasset det til min oppgave. I
hovedundersøkelsen ser jeg både på generelle
metoder (Polyas modell) og på spesifikke
problemløsningsheuristikker. For å analysere
forekomsten av problemløsningsmetoder i
eksemplene i de undersøkte matematikklærebøkene
brukte jeg et analyseverktøy utviklet av
doktorgradsstudenten Tom Rune Kongelf ved
Universitetet i Agder.
Analysen av både den norske læreplanen og den
russiske læreplanen slår fast at elevene burde
lære problemløsning og problemløsningsmetoder.
Tross det viser hovedresultatene at det ikke
finnes generell innføring i problemløsning i de
norske og de russiske matematikklærebøkene.
Verken de norske eller de russiske lærebøkene
presenterer generelle metoder og spesifikke
problemløsningsheuristikker i tilstrekkelig
grad. Oppgaven konkluderte med at det finnes
både fellestrekk og ulikhetstrekk mellom de
norske og de russiske lærebøkene når det
gjelder problemløsning. Men samtidig ble det
oppdaget mange interessante funn ved siden av
hovedanalysen.
Sun, 17 Jan 2016 00:00:00 GMThttp://hdl.handle.net/1956/158112016-01-17T00:00:00ZKalmanfilteret anvendt på en problemstilling innen seismikkhttp://hdl.handle.net/1956/15810
Kalmanfilteret anvendt på en problemstilling innen seismikk
Kolnes, Marianne Thorsnæs
Master thesis
I denne oppgaven er målet å estimere, samt predikere, posisjonen til seimikk-kabler sekvensielt. Dessuten blir det studert om målinger fra et akselerometer, festet på seismikk-kabelen, gir sikere inferens om posisjonen.
Wed, 25 Nov 2009 00:00:00 GMThttp://hdl.handle.net/1956/158102009-11-25T00:00:00ZSurface reconstruction Using Probability Based Photo-Consistencyhttp://hdl.handle.net/1956/15808
Surface reconstruction Using Probability Based Photo-Consistency
Kjørstad, Åsmund
Master thesis
We reconstruct the surface of an object seen from many different angles. We construct an energy functional by using visibility constraints and photo-consistency. This is then solved by the piecewise constant level set method.
Fri, 04 Dec 2009 00:00:00 GMThttp://hdl.handle.net/1956/158082009-12-04T00:00:00ZPolish Translation and Validation of the Tinnitus Handicap Inventory and the Tinnitus Functional Indexhttp://hdl.handle.net/1956/15769
Polish Translation and Validation of the Tinnitus Handicap Inventory and the Tinnitus Functional Index
Wrzosek, Małgorzata; Eugeniusz, Szymiec; Wiesława, Klemens; Kotyło, Piotr; Schlee, Winfried; Modrzynska, Małgorzata; Agnieszka, Lang-Małecka; Anna, Preis; Bulla, Jan
Journal article
<p>Objective: The need for validated measures enabling clinicians to classify tinnitus patients according to the severity of tinnitus and screen the progress of therapies in our country led us to translate into Polish and to validate two tinnitus questionnaires, namely the Tinnitus Handicap Inventory (THI) and the Tinnitus Functional Index (TFI).</p>
<p>Design: The original English versions of the questionnaires were translated into Polish and translated back to English by three independent translators. These versions were then finalized by the authors into a Polish THI (THI-Pl) and a Polish TFI (TFI-Pl). Participants from three laryngological centers in Poland anonymously answered the THIPl (N = 98) and the TFI-Pl (N = 108) in addition to the Polish versions of the Center for Epidemiologic Studies Depression Scale as a measure of self-perceived level of depression, and the Satisfaction With Life Scale to assess self-perceived quality of life. Both were used to determine discriminant validity. Two Visual Analog Scales were used to measure tinnitus annoyance and tinnitus loudness in order to determine convergent validity.</p>
<p>Results: Similar to the original version of the THI, the THI-Pl showed a high internal consistency (Cronbach&rsquo;s &alpha; = 0.93). The exploratory factor analysis revealed that the questionnaire has a three-factorial structure that does not correspond to the original division for functional, catastrophic, and emotional subscales. Convergent and discriminant validities were confirmed. The TFI-Pl showed high internal consistency (Cronbach&rsquo;s &alpha; = 0.96) with the reliability ranging from 0.82 to 0.95 for its different subscales. Factor analysis confirmed an eight-factorial structure with factors assigning all items to appropriate subscales reported in the original version of the questionnaire. Discriminant and convergent validities were also confirmed for the TFI-Pl.</p>
<p>Conclusion: We translated and validated the Polish versions of the THI and the TFI to make them suitable for clinical use in Poland.</p>
Tue, 29 Nov 2016 00:00:00 GMThttp://hdl.handle.net/1956/157692016-11-29T00:00:00ZExact solutions for shoaling waves in shallow waterhttp://hdl.handle.net/1956/15747
Exact solutions for shoaling waves in shallow water
Bjørnestad, Maria
Master thesis
The dynamics of shallow-water waves at the surface of an inviscid and incompressible fluid over a background shear flow approaching a sloping beach are investigated. First, we derive the nonlinear shallow-water equations in the presence of both background shear flow and a sloping beach. In this case, the hyperbolic shallow-water equations are not reducible and is it not straightforward to find the Riemann invariants. However, using intuition gained from the case of a shear flow over a flat bed, Riemann invariants can nevertheless be found. The Riemann invariants provide a proper hodograph transformation which is combined with several additional changes of variables to put the equations in linear form. This linear equation can be solved using the method of separation of variables. In this way, we are able to find exact solutions which give us a prediction of the shoaling process and of the development of the waterline (run-up). Our work is inspired by the method of Carrier and Greenspan [4]. Therefore, a careful description of the Carrier-Greenspan method is presented first.
Wed, 22 Mar 2017 00:00:00 GMThttp://hdl.handle.net/1956/157472017-03-22T00:00:00ZPDE Based Algorithms for Smooth Watershedshttp://hdl.handle.net/1956/15684
PDE Based Algorithms for Smooth Watersheds
Hodneland, Erlend; Tai, Xue-Cheng; Kalisch, Henrik
Journal article
Watershed segmentation is useful for a number of image segmentation problems with a wide range of practical applications. Traditionally, the tracking of the immersion front is done by applying a fast sorting algorithm. In this work, we explore a continuous approach based on a geometric description of the immersion front which gives rise to a partial differential equation. The main advantage of using a partial differential equation to track the immersion front is that the method becomes versatile and may easily be stabilized by introducing regularization terms. Coupling the geometric approach with a proper “merging strategy” creates a robust algorithm which minimizes over- and under-segmentation even without predefined markers. Since reliable markers defined prior to segmentation can be difficult to construct automatically for various reasons, being able to treat marker-free situations is a major advantage of the proposed method over earlier watershed formulations. The motivation for the methods developed in this paper is taken from high-throughput screening of cells. A fully automated segmentation of single cells enables the extraction of cell properties from large data sets, which can provide substantial insight into a biological model system. Applying smoothing to the boundaries can improve the accuracy in many image analysis tasks requiring a precise delineation of the plasma membrane of the cell. The proposed segmentation method is applied to real images containing fluorescently labeled cells, and the experimental results show that our implementation is robust and reliable for a variety of challenging segmentation tasks.
Fri, 01 Apr 2016 00:00:00 GMThttp://hdl.handle.net/1956/156842016-04-01T00:00:00ZStable cell-centered finite volume discretization for biot equationshttp://hdl.handle.net/1956/15633
Stable cell-centered finite volume discretization for biot equations
Nordbotten, Jan Martin
Journal article
In this paper we discuss a new discretization for the Biot equations. The discretization
treats the coupled system of deformation and flow directly, as opposed to combining
discretizations for the two separate subproblems. The coupled discretization has the following key
properties, the combination of which is novel: (1) The variables for the pressure and displacement
are co-located and are as sparse as possible (e.g., one displacement vector and one scalar pressure
per cell center). (2) With locally computable restrictions on grid types, the discretization is stable
with respect to the limits of incompressible fluid and small time-steps. (3) No artificial stabilization
term has been introduced. Furthermore, due to the finite volume structure embedded in the
discretization, explicit local expressions for both momentum-balancing forces and mass-conservative
fluid fluxes are available. We prove stability of the proposed method with respect to all relevant
limits. Together with consistency, this proves convergence of the method. Finally, we give numerical
examples verifying both the analysis and the convergence of the method.
Tue, 29 Mar 2016 00:00:00 GMThttp://hdl.handle.net/1956/156332016-03-29T00:00:00Z"Jeg har aldri jobbet så mye som nå, og så får jeg en treer, liksom?" - En studie av elevers læringsstrategier og overgangsvansker i møte med Matematikk 1Thttp://hdl.handle.net/1956/15625
"Jeg har aldri jobbet så mye som nå, og så får jeg en treer, liksom?" - En studie av elevers læringsstrategier og overgangsvansker i møte med Matematikk 1T
Bratlie, Aleksander Willatz
Master thesis
Denne masteroppgaven handler om de elevene som ved
overgangen fra ungdomsskolen til videregående
skole opplever møtet med matematikkurset 1T som
spesielt vanskelig. Elevene jeg studerer har på
papiret et godt utgangspunkt og høy måloppnåelse i
matematikk fra ungdomsskolen, og de er motiverte
for å begynne på videregående skole og lære mer
matematikk. Men i møtet med 1T opplever de en
sterk nedgang i karakterer uten at de forstår
hvorfor, og de føler at de ikke er i stand til å
gjøre noe med dette selv. Jeg har undersøkt
læringsstrategier blant disse elevene, og studert
om det er samsvar mellom elevenes bruk av
læringsstrategier og deres besvarelse av et utvalg
oppgaver på terminprøven og under intervju.
Dette er et mixed methods-studie med hovedvekt på
det kvalitative. Hensikten med studien var å
avdekke spesifikke læringsstrategier og eventuelle
lærevansker hos en utvalgt gruppe fokuselever. Jeg
gjennomførte intervjuer med i alt fem frivillige
informanter. Gjennom spørsmål om mestring og
motivasjon, læring og læringsstrategier, og
løsning av et utvalg oppgaver, gav intervjuene
interessant informasjon om elevenes forhold til
matematikk og læringsstrategier. Gjennom analyser
av transkripsjoner og oppgaveløsning, avdekket jeg
at samtlige informanter hadde et begrenset forhold
til læringsstrategier i matematikk. Informantene
benyttet seg i stor grad av enkle strategier som
tidligere hadde vist seg å fungere tilstrekkelig.
De hadde vanskelig for å utvikle nye, effektive
strategier på videregående skole og samtidig henge
med i den relativt raske progresjonen i 1T-kurset,
som for mange av informantene var uvant og
problematisk. Resultatene kan ses i lys av
relevant teori, om at det er sammenheng mellom
elevenes bruk av læringsstrategier i matematikk,
deres forståelse av hva matematikk er, og deres
evne til å anvende tillært kunnskap i nye
sammenhenger.
Sun, 20 Nov 2016 00:00:00 GMThttp://hdl.handle.net/1956/156252016-11-20T00:00:00ZMultiscale analysis of selected problems in fluid dynamicshttp://hdl.handle.net/1956/15563
Multiscale analysis of selected problems in fluid dynamics
Alyaev, Sergey
Doctoral thesis
<p>The world around us is inherently multiscale. The variety of different models depending on the reference scale is particularly diverse in problems of fluid-solid systems. For those systems, the geometry, effective parameters and even equations vary between scales. Normally, on coarser scales the fundamental laws of physics are not sufficient to provide a closed-form model. This called for experiment-based closures, which often had limited domain of applicability. An alternative approach enabled by recent developments is so-called multiscale methods. By considering the two scales simultaneously, multiscale methods do not rely on experimental closures and therefore lay a physics-based foundation for engineering problems.</p>
<p>In this work, we attempt to separate multiscale methods into larger classes based on a flowchart that contains questions about the structure of problems, for which they are designed. Analysis and methods, presented in the thesis, aim to fill the gaps in inter-disciplinary knowledge across several classes of multiscale methods. We conduct fit-for-purpose analysis of several selected multiscale problems in fluid dynamics: nonlinear single phase flows in porous media, fractal structures formation in freezing brine, particle-particle interaction affected by capillary bridging.</p>
<p>The first problem is addressed by the control volume heterogeneous multiscale method (CVHMM). By coupling mass conservation with Navier-Stokes equations on the pore scale, the method captures nontrivial coarse-scale effects, which are not handled by standard single-scale models. To demonstrate the robustness of the presented methods, a multiscale convergence analysis and a derivation of a priori error estimates are developed for the method.</p>
<p>For the second problem, simplifications and consequent self-similar analysis identifies compact and fractal regimes of ice formation. Moreover, combining the insights obtained from numerical and analytical solutions we introduce an empirical model that can qualitatively predict properties of the formed ice.</p>
<p>For the third problem, we derive a new semi-analytical solution for particle interactions during viscous bridging. The solution provides basis for more efficient simulation of suspended flows of particles.</p>
Fri, 03 Feb 2017 00:00:00 GMThttp://hdl.handle.net/1956/155632017-02-03T00:00:00ZAnalysis of Control Volume Heterogeneous Multiscale Methods for Single Phase Flow in Porous Mediahttp://hdl.handle.net/1956/15562
Analysis of Control Volume Heterogeneous Multiscale Methods for Single Phase Flow in Porous Media
Alyaev, Sergey; Keilegarden, Eirik; Nordbotten, Jan Martin
Journal article
The standard approximation for the flow-pressure relationship in porous media is Darcy's law that was originally derived for infiltration of water in fine homogeneous sands. Ever since there have been numerous attempts to generalize it for handling more complex flows. Those include upscaling of standard continuum mechanics flow equations from the fine scale. In this work we present a heterogeneous multiscale method that utilizes fine scale information directly to solve problems for general single phase flow on the Darcy scale. On the coarse scale it only assumes mathematically justified conservation of mass on control volumes, that is, no phenomenological Darcy-type relationship for velocity is presumed. The fluid fluxes are instead provided by a fine scale Navier--Stokes mixed finite element solver. This work also considers several choices of quadrature for data estimation in the multiscale method and compares them. We prove that for an essentially linear regime, when the fine scale is governed by Stokes flow, our method converges to a rigorously derived homogenization solution---Darcy's law. Moreover the method gives the flexibility to solve problems with faster nonlinear flow regimes that is important in a number of applications, such as flows that may occur near wells and in fractured regions in subsurface. Those flows are also common for industrial and near surface porous media. The numerical examples presented in the paper verify the estimate and emphasize the importance of good data estimation.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/1956/155622014-01-01T00:00:00ZMultiscale Simulation of Non-Darcy Flowshttp://hdl.handle.net/1956/15561
Multiscale Simulation of Non-Darcy Flows
Alyaev, Sergey; Keilegarden, Eirik; Nordbotten, Jan Martin
Conference object
In this work we present control volume multiscale methods which address problems on the interaction between pore scale and Darcy scale. For the case when linear equations govern the ow on the pore scale our solution converges to semi-analytical solution. Moreover, the method also extends to non-linear problems governed by Navier- Stokes equations. We show numerical results illustrating the applicability of the method.
XIX International Conference on Water Resources, CMWR 2012
Fri, 01 Jun 2012 00:00:00 GMThttp://hdl.handle.net/1956/155612012-06-01T00:00:00ZA simulation workflow for large-scale CO2 storage in the Norwegian North Seahttp://hdl.handle.net/1956/15482
A simulation workflow for large-scale CO2 storage in the Norwegian North Sea
Lie, Knut-Andreas; Nilsen, Halvor Møll; Andersen, Odd; Møyner, Olav
Journal article
Large-scale CO2 injection problems have revived the interest in simple models, like percolation and vertically-averaged models, for simulating
fluid flow in reservoirs and aquifers. A series of such models have been collected and
implemented together with standard reservoir simulation capabilities in a
high-level scripting language as part of the open-source MATLAB Reservoir
Simulation Toolbox (MRST) to give a set of simulation methods of increas-
ing computational complexity. Herein, we outline the methods and discuss
how they can be combined to create a flexible tool-chain for investigating
CO2 storage on a scale that would have significant impact on European
CO2 emissions. In particular, we discuss geometrical methods for identify-
ing structural traps, percolation-type methods for identifying potential spill
paths, and vertical-equilibrium methods that can effciently simulate struc-
tural, residual, and solubility trapping in a thousand-year perspective. The
utility of the overall workflow is demonstrated using real-life depth and thick-
ness maps of two geological formations from the recent CO2 Storage Atlas of
the Norwegian North Sea.
Wed, 01 Jun 2016 00:00:00 GMThttp://hdl.handle.net/1956/154822016-06-01T00:00:00ZReexamining CO2 Storage Capacity and Utilization of the Utsira Formationhttp://hdl.handle.net/1956/15480
Reexamining CO2 Storage Capacity and Utilization of the Utsira Formation
Andersen, Odd; Nilsen, Halvor Møll; Lie, Knut-Andreas
Conference object
In this work we provide estimates of CO2 storage capacity of the Utsira Formation using the recently provided datasets from the Norwegian Petroleum Directorate, taking CO2 density variation into account. We also investigate strategies on how to realize as much as possible of this potential in large-scale injection scenarios. We base our study on the assumption that the limiting factor for CO2 storage at Utsira is the efficient use of available trapping mechanisms, with pressure buildup being of secondary concern. We consider the trapping mechanisms considered most important for the medium-to-long term (structural, residual and dissolution trapping), using a combination of modeling tools based on the Matlab Reservoir Simulation Toolbox (MRST).
Presented at: ECMOR XIV - 14th European conference on the mathematics of oil recovery, Catania, Italy, 8.-11. September 2014
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/1956/154802014-01-01T00:00:00ZSimplified models for numerical simulation of geological CO2 storagehttp://hdl.handle.net/1956/15477
Simplified models for numerical simulation of geological CO2 storage
Andersen, Odd
Doctoral thesis
<p>Carbon capture and storage (CCS) is a proposed strategy to reduce global emissions
of greenhouse-gases. The basic principle is to capture CO2 from power generation or
other industrial activities and inject it into deep geological formations for permanent
storage. CCS is considered practically indispensable by the Intergovernmental Panel
on Climate Change (IPCC) in order to reach internationally agreed climate targets. It
can be understood as a bridge technology intended to limit emissions from fossil-fuel
based economic activities while working toward the longer-term goal of a sustainable
energy system.</p>
<p>Since the purpose of CCS is to permanently prevent large quantities of CO2 from
entering the carbon cycle, the practical storage capacity and long-term safety of candidate
storage sites are important questions to address. As for other industrial subsurface
operations, numerical simulations based on mathematical models of the involved
physics play a key role in helping us understand the processes taking place underground.
However, existing industrial simulators for 3D subsurface multi-phase flow
are typically developed for the support of hydrocarbon production. Such simulators
are by nature limited in their ability to handle problems at the very large spatial and
temporal scales that must be taken into account when investigating CO2 storage issues.
To properly address the full range of questions related to CO2 injection and migration,
a variety of mathematical models of different complexity is needed, ranging from detailed
multi-physics models describing local conditions around the injection sites, to
simplified or mathematically upscaled descriptions capable of modeling developments
at much larger spatial scales and timeframes.</p>
<p>This thesis addresses the development, analysis and efficient implementation of simplified
mathematical models specially designed to address questions related to longterm
CO2 storage. One class of such models enables computationally efficient longterm
simulations based on the assumption of vertical equilibrium (VE). Under this assumption,
vertical flow in the storage formation is neglected, which allows for reducing
the dimensions of the governing equations and corresponding simulation domain from
three to two, while still preserving important 3D effects. A different numerical approach
is based on analysis of storage site geometry, and provides a near-instant way of
predicting long-term migration pattern and assessing trapping capacity. Together, these
methods can be applied as parts of larger workflows set up to address more complex
questions related to CO2 storage.</p>
<p>The work presented in this thesis contributes to the field of mathematical modeling
of CO2 storage with several new developments, including:
<ul>
<li>mathematical derivation and inclusion of additional physical effects into the VE
modeling framework (compressibility, geomechanics, hysteresis models), and assessment of the impacts;</li>
<li>efficient algorithms for spill-point analysis of the storage site caprock, and evaluating
their applicability for CO2 migration prediction, trapping capacity estimation
and well placement support, based on testing on real/realistic datasets;</li>
<li>robust implementation of vertical-equilibrium models in a fully-implicit, blackoil
framework, combining most of the modeling capabilities that have been published
for these types of models (dissolution, residual trapping, capillary pressure,
caprock rugosity, and dynamic fluid properties derived from equations of state),
followed by testing and validation based on real aquifer models from the North
Sea;</li>
<li>use of the aforementioned approaches in combination with gradient-based nonlinear
optimization methods to identify practical injection scenarios that maximize
stored CO2 while minimizing migration out of the target formation.</li>
</ul>
</p>
<p>In the spirit of promoting reproducible computational research [94], most of the
computer code underlying the results presented has been made freely available as open
software in the form of a separate module, MRST-co2lab, to the MATLAB Reservoir
Simulation Toolbox (MRST), developed, maintained and published by the Computational
Geosciences group of SINTEF ICT, Department of Applied Mathematics. The exception
is the work on geomechanics, which has not yet been made part of the public code,
but is in the pipeline for a future release.</p>
<p>The introduction to this thesis is written as a tutorial that introduces some of the
basic theory underlying the papers, and also demonstrates the practical use of the software
by gradually outlining a full code example based on a publicly available dataset
of the Johansen formation.</p>
Fri, 03 Feb 2017 00:00:00 GMThttp://hdl.handle.net/1956/154772017-02-03T00:00:00ZInterest rate models in Solvency IIhttp://hdl.handle.net/1956/15421
Interest rate models in Solvency II
Sivertsen, Kristine
Master thesis
The best estimate of liabilities is important in
the Solvency II framework. The
best estimate of liabilities should be
probability weighted average of future cash
flows discounted to its present value. Life
insurance companies need stochastic
models to produce future paths for interest
rates, bond returns and currency.
These paths should be risk-neutral, meaning that
interest rate models is important
to consider in the Solvency II framework. In this
thesis we have studied
three different interest rate models, namely; the
Hull-White extended Vasicek
model, the CIR++ model and the G2++ model.
We calibrated our interest rate models to the
same historical data and generated
10 000 simulations based on the yield curve and
the parameter estimations.
Based on the interest rates simulated we
presented a synthetic example for calculating
the best estimate of liabilities. In this
example, the duration of the
liabilities turned out to be an important factor.
Mon, 21 Nov 2016 00:00:00 GMThttp://hdl.handle.net/1956/154212016-11-21T00:00:00ZOmvendt undervisning i matematikk. En kvalitativ studie av hvordan matematikkfaget endrer seg ved bruk av omvendt undervisning, og hva lærere og elever må passe på for å sikre kvaliteten på opplæringen i faget gjennom disse endringene.http://hdl.handle.net/1956/15415
Omvendt undervisning i matematikk. En kvalitativ studie av hvordan matematikkfaget endrer seg ved bruk av omvendt undervisning, og hva lærere og elever må passe på for å sikre kvaliteten på opplæringen i faget gjennom disse endringene.
Myhr, Geir
Master thesis
Problemstillingen for studien er: Hvordan endrer matematikkfaget seg ved bruk av omvendt undervisning, og hvilke faktorer må en som lærer, og elev, passe på for å sikre kvalitet på opplæringen gjennom disse endringene? Studiens hovedfunn: - Elevene opplever å være bedre forberedt til timene - innholdet i timene består i hovedsak av samtale og diskusjon rundt matematiske tema, og variert arbeid med oppgaver - Lærerens rolle har endret seg fra å være en som presenterer informasjon, til en som i all hovedsak er veileder for elevene i deres læring. Med bakgrunn i studiens hovedfunn, gjøres noen betrakninger rundt elevenes ansvar og undervisningens kvalitet.
Mon, 21 Nov 2016 00:00:00 GMThttp://hdl.handle.net/1956/154152016-11-21T00:00:00ZMisoppfatninger innen betinget sannsynlighet for elever i den videregående skolehttp://hdl.handle.net/1956/15409
Misoppfatninger innen betinget sannsynlighet for elever i den videregående skole
Holst, Erik
Master thesis
Masteroppgaven handler om ulike misoppfatninger elever på videregående skole har innen emnet betinget sannsynlighet. I teorien belyses misoppfatninger, feilslutninger og heuristikker med hovedvekt på betinget sannsynlighet. I oppgaven har jeg brukt mixed method, en blanding mellom kvantitativ og kvalitativ metode. Til den kvantitative delen har jeg brukt oppgaver innen betinget sannsynlighet utarbeidet av Diaz og Fuente. De kaller denne samlingen av oppgaver for CPR (Conditional Probability Reasoning). Jeg bruker de samme oppgavene i den kvalitative delen av masteroppgaven i oppgavebasert intervju. Jeg sammenligner mine resultat med resultat fra CPR, og studerer graden av ulike misoppfatninger, relatert til grupper som har hatt eller ikke har hatt undervisning i emnet. Hvilke misoppfatninger som reduseres, består eller øker, diskuteres før sammendrag, konklusjon og implikasjon.
Tue, 20 Sep 2016 00:00:00 GMThttp://hdl.handle.net/1956/154092016-09-20T00:00:00ZBruk av matematikkens historie i matematikkundervisningen - En kilde til motivasjon?http://hdl.handle.net/1956/15408
Bruk av matematikkens historie i matematikkundervisningen - En kilde til motivasjon?
Kristiansen, Jarl Harald
Master thesis
Denne masteroppgaven tar for seg bruk av
matematikkens historie i matematikkundervisning.
Jeg har selv en bakgrunn som matematikklærer i
videregående skole gjennom drøyt 15 år, og for meg
personlig har matematikkens historie vært en
inspirasjonskilde og også bidratt til økt
forståelse av matematikken. Jeg ønsket derfor å
undersøke om det var mulig å bruke matematikkens
historie i matematikkundervisningen og om dette
hadde positiv effekt på elevenes holdning til
matematikkfaget.
Jeg har utarbeidet et undervisningsopplegg for
elever i den videregående skolen, bestående av tre
oppgavesett som alle er basert på historiske
kilder. Undervisningen ble gjennomført på en
skoledag for elever på studiespesialiserende
programområde Vg2, etter at disse var ferdige med
sine eksamener. Da utvalget av elever som deltok i
forsøket hadde valgt ulike matematikkfag, valgte
jeg å basere oppgavene på tema som ikke pensum i
noen av matematikkfagene i den videregående skolen.
Jeg har først og fremst ønsket å se på om dette er
en måte å motivere elevene for matematikk. Elevene
har derfor svart på en spørreundersøkelse om
holdninger til matematikkfaget både før og etter
gjennomføringen av undervisningopplegget. I tillegg
ble det gjennomført gruppeintervju med noen av
elevene i etterkant av undervisningen.
Spørreundersøkelse og intervju danner grunnlaget
for analysen, sammen med de observasjonene jeg
gjorde den dagen jeg var sammen med elevene.
Spørreundersøkelsen bidrar med kvantitative
resultater, men er basert på få respondenter.
Påstandene i spørreundersøkelsen er ordnet i fire
ulike kategorier; underholdningsverdi, motivasjon,
trygghet og verdi. For tre av de fire variablene
som er undersøkt i spørreundersøkelsen finner jeg
ingen signifikante forskjeller, mens en av
variablene, underholdningsverdien av matematikk,
har en signifikant økning. Både de kvalitative og
de kvantitative resultatene stemmer bra overens, og
jeg finner lignende resultater i andre studier.
Tilbakemeldingene fra elevene er positive både på
innhold og utforming av oppgavene, men de gir også
tydelig uttrykk for at fagene allerede er presset
på tid, så faglig aktivitet utenom pensum vil de
ikke bruke tid på. Det betyr at å gjennomføre et
slikt opplegg i en vanlig klasse krever en del
forarbeid fra lærerens side, da oppgaver og teori
som kan knyttes til pensum ikke nødvendigvis er
lett tilgjengelig.
Mon, 21 Nov 2016 00:00:00 GMThttp://hdl.handle.net/1956/154082016-11-21T00:00:00ZA Comparison of Two Numerical Models for Flow in Fractured Porous Media and the Impact of Fracture Intersection Cell Removalhttp://hdl.handle.net/1956/15407
A Comparison of Two Numerical Models for Flow in Fractured Porous Media and the Impact of Fracture Intersection Cell Removal
Stefansson, Ivar
Master thesis
The simulation of flow and transport in fractured porous media is crucial to improve
understanding and viability of geothermal energy. It is also challenging, due to complex
geometries and large scale differences. In this thesis, we compare two numerical methods
developed to meet these needs; a Finite Volume method and a Mixed Finite Element method.
We analyse their behaviour based on a collection of test cases. In addition to studying the
produced pressure solutions, we perform transport simulations to reveal local behaviour through
accumulation.
We take particular interest in the intersections of fractures and the possibilities in removing the
intersection cells from the spatial discretization, improving condition numbers and relaxing time
step restrictions. We identify some dangers in eliminating the cells before the flow is solved, and
instead propose a procedure to be applied between the flow simulation and the transport
simulation.
Mon, 21 Nov 2016 00:00:00 GMThttp://hdl.handle.net/1956/154072016-11-21T00:00:00ZMultivariate and conditional density estimation using local Gaussian approximationshttp://hdl.handle.net/1956/15333
Multivariate and conditional density estimation using local Gaussian approximations
Otneim, Håkon
Doctoral thesis
<p>Paper 1 ”Bias and bandwidth for local likelihood density estimation”:
A local likelihood density estimator is shown to have asymptotic bias depending on
the dimension of the local parameterization. Comparing with kernel estimation it is
demonstrated using a variety of bandwidths that we may obtain as good, and potentially
even better estimates using local likelihood. Boundary effects are also examined.</p>
<p>Paper 2 ”The locally Gaussian density estimator for multivariate data”:
It is well known that the Curse of Dimensionality causes the standard Kernel Density
Estimator to break down quickly as the number of variables increases. In non-parametric
regression, this effect is relieved in various ways, for example by assuming additivity
or some other simplifying structure on the interaction between variables. This paper
presents the Locally Gaussian Density Estimator (LGDE), which introduces a similar
idea to the problem of density estimation.
The LGDE is a new method for the non-parametric estimation of multivariate probability
density functions. It is based on preliminary transformations of the marginal
observation vectors towards standard normality, and a simplified local likelihood fit of
the resulting distribution with standard normal marginals. The LGDE is introduced,
and asymptotic theory is derived. In particular, it is shown that the LGDE converges
at a speed that does not depend on the dimension. Examples using real and simulated
data confirm that the new estimator performs very well on finite sample sizes.</p>
<p>Paper 3 ”Non-parametric estimation of conditional density functions: A new method”:
Let X = (X1, . . . , Xp) be a stochastic vector having joint density function fX(x) with
partitions X1 = (X1, . . . , Xk) and X2 = (Xk+1, . . . , Xp). A new method for estimating
the conditional density function of X1 given X2 is presented. It is based on locally
Gaussian approximations, but simplified in order to tackle the curse of dimensionality in
multivariate applications, where both response and explanatory variables can be vectors.
We compare our method to some available competitors, and the error of approximation is
shown to be small in a series of examples using real and simulated data, and the estimator
is shown to be particularly robust against noise caused by independent variables. We
also present examples of practical applications of our conditional density estmator in the
analysis of time series. Typical values for k in our examples are 1 and 2, and we include
simulation experiments with values of p up to 6. Large sample theory is established
under a strong mixing condition.</p>
Fri, 23 Sep 2016 00:00:00 GMThttp://hdl.handle.net/1956/153332016-09-23T00:00:00ZSimulating spatial and temporal varying CO2 signals from sources at the seafloor to help designing risk-based monitoring programshttp://hdl.handle.net/1956/15290
Simulating spatial and temporal varying CO2 signals from sources at the seafloor to help designing risk-based monitoring programs
Ali, Alfatih Omer Mohammed Ahmed; Frøysa, Håvard G; Avlesen, Helge; Alendal, Guttorm
Journal article
Risk-based monitoring requires quantification of the probability of the design to detect the potentially adverse events. A component in designing the monitoring program will be to predict the varying signal caused by an event, here detection of a gas seep through the seafloor from an unknown location. The Bergen Ocean Model (BOM) is used to simulate dispersion of CO2 leaking from different locations in the North Sea, focusing on temporal and spatial variability of the CO2 concentration. It is shown that the statistical footprint depends on seep location and that this will have to be accounted for in designing a network of sensors with highest probability of detecting a seep. As a consequence, heterogeneous probabilistic predictions of CO2 footprints should be available to subsea geological CO2 storage projects in order to meet regulations.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/1956/152902016-01-01T00:00:00ZIterative schemes for solving coupled, non-linear flow and transport in porous mediahttp://hdl.handle.net/1956/15288
Iterative schemes for solving coupled, non-linear flow and transport in porous media
Illiano, Davide
Master thesis
In this thesis we compare different iterative approaches for solving the non-linear, coupled multiphase flow and reactive transport in porous media. Especially, we consider two-phase flow and one a-half phase flow (modeled by Richards equation) coupled with an one component transport equation. Implicit and explicit iterative schemes will be compared in terms of efficiency, robustness and accuracy. The best approach seems to be a fully implicit scheme, which is a slightly modified variant of the classical splitting iterative scheme for coupled equations. We concentrate on three linearization methods: L-scheme, Modified Picard and Newton. We implement them in Matlab and tested on both an academic example, built from a manufactured analytical solution and on a realistic problem, the Salinity Problem. In the first part of the thesis we will also briefly study the generic two-phase flow plus transport equation in porous media, presenting some of the most common solving algorithms such as the IMPES method and its Fully Implicit reformulation.
Sat, 24 Dec 2016 00:00:00 GMThttp://hdl.handle.net/1956/152882016-12-24T00:00:00ZRobust simulation of sharp-interface models for fast estimation of CO2 trapping capacity in large-scale aquifer systemshttp://hdl.handle.net/1956/15242
Robust simulation of sharp-interface models for fast estimation of CO2 trapping capacity in large-scale aquifer systems
Nilsen, Halvor Møll; Lie, Knut-Andreas; Andersen, Odd
Journal article
Modeling geological carbon storage represents a new and substantial challenge for the subsurface geosciences. To increase understanding and make good engineering decisions, containment processes and large-scale storage operations must be simulated in a thousand-year perspective. Large differences in spatial and temporal scales makes it prohibitively expensive to compute the fate of injected CO2 using traditional 3D simulators. Instead, accurate forecast can be computed using simplified models that are adapted to the specific setting of the buoyancy-driven migration of the light fluid phase.
This paper presents a family of vertically integrated models for studying the combined large-scale and long-term effects of structural, residual, and solubility trapping of CO2. The models are based on an assumption of a sharp interface separating CO2 and brine and can provide a detailed inventory of the injected CO2 volumes over periods of thousands of years within reasonable computational time. To be compatible with simulation tools used in industry, the models are formulated in a black-oil framework. The models are implemented in MRST-co2lab, which is an open community software developed especially to study and optimize large-scale, long-term geological storage of CO2. The resulting simulators are fully implicit and handle input from standard geomodeling tools.
Mon, 01 Feb 2016 00:00:00 GMThttp://hdl.handle.net/1956/152422016-02-01T00:00:00ZAdaptive Multiscale Methods Based on A Posteriori Error Estimateshttp://hdl.handle.net/1956/13151
Adaptive Multiscale Methods Based on A Posteriori Error Estimates
Alyaev, Sergey
Master thesis
We give an overview of different methods for solving highly heterogeneous
elliptic problems with multiscale structure and no intuitive scale separation.
We compare different finite element variational multiscale methods and prove
equivalence between the methods proposed by Larson et al. [LM07] and Nolen
et al [NPP08]. We also discuss properties of different multiscale methods
depending on the choice of scale separation and ways to represent the fine
scale correction. Additionally, in this work we give an overview of a posteriori
error estimates for the finite element method as well as newly proposed by
Larson et al. estimates for the variational multiscale method [LM07]. As an
illustration of the theory we show our numerical results for using theoretical
estimates to construct adaptive algorithms: adaptive refinement of finite
elements and adaptive overlap control for variational multiscale methods in
the formulation of Nolen et al. There is no known implementation of the
latter published at the moment.
Tue, 01 Jun 2010 00:00:00 GMThttp://hdl.handle.net/1956/131512010-06-01T00:00:00ZFully-implicit simulation of vertical-equilibrium models with hysteresis and capillary fringehttp://hdl.handle.net/1956/13123
Fully-implicit simulation of vertical-equilibrium models with hysteresis and capillary fringe
Nilsen, Halvor Møll; Lie, Knut-Andreas; Andersen, Odd
Journal article
Geological carbon storage represents a new and substantial challenge for the subsurface geosciences. To increase understanding and make good engineering decisions, containment processes and large-scale storage operations must be simulated in a thousand year perspective.
A hierarchy of models of increasing computational complexity for analysis and simulation of large-scale \co storage has been implemented as a separate module of the open-source Matlab Reservoir Simulation Toolbox (MRST). This paper describes a general family of two-scale models available in this module. The models consist of two-dimensional flow equations formulated in terms of effective quantities obtained from hydrostatic reconstructions of vertical pressure and saturation distributions. The corresponding formulation is fully implicit and is the first to give a mass-conservative treatment and include general (non-linearized) CO2 properties. In particular, the models account for compressibility, dissolution, and hysteresis effects in the fine-scale capillary and relative permeability functions and can be used to accurately and efficiently study the combined large-scale and long-term effects of structural, residual, and solubility trapping.
Mon, 01 Feb 2016 00:00:00 GMThttp://hdl.handle.net/1956/131232016-02-01T00:00:00ZEr konsum av melk og egg assosiert med risiko for kreft? Foreløpige resultater fra en prospektiv undersøkelsehttp://hdl.handle.net/1956/12914
Er konsum av melk og egg assosiert med risiko for kreft? Foreløpige resultater fra en prospektiv undersøkelse
Albrektsen, Grethe; Heuch, Ivar; Jacobsen, Bjarne K.; Kvåle, Gunnar
Journal article
Wed, 01 Jan 1997 00:00:00 GMThttp://hdl.handle.net/1956/129141997-01-01T00:00:00ZA study on iterative methods for Richards' equationhttp://hdl.handle.net/1956/12738
A study on iterative methods for Richards' equation
List, Florian; Radu, Adrian Florin
Journal article
This work concerns linearization methods for efficiently solving the Richards equation, a degenerate elliptic-parabolic equation which models flow in saturated/unsaturated porous media. The discretization of Richards’ equation is based on backward Euler in time and Galerkin finite elements in space. The most valuable linearization schemes for Richards’ equation, i.e. the Newton method, the Picard method, the Picard/Newton method and the L-scheme are presented and their performance is comparatively studied. The convergence, the computational time and the condition numbers for the underlying linear systems are recorded. The convergence of the L-scheme is theoretically proved and the convergence of the other methods is discussed. A new scheme is proposed, the L-scheme/Newton method which is more robust and quadratically convergent. The linearization methods are tested on illustrative numerical examples.
Fri, 01 Apr 2016 00:00:00 GMThttp://hdl.handle.net/1956/127382016-04-01T00:00:00ZTokamerarekonstruksjon og Gröbnerbasarhttp://hdl.handle.net/1956/12658
Tokamerarekonstruksjon og Gröbnerbasar
Slinde, Kristine Njøs
Master thesis
I denne oppgåva skal me sjå på korleis ein skal gå fram for å rekonstruere ei scene frå to bilete. Dette kan brukast når ein skal evaluere satellittbilete og gjere ei kalibrering av kamera. Me skal sjå på geometrien og matematikken bak tilfellet der ein har eit punkt i rommet, som blir avbilda i to bilete. Dette punktet vil bli avbilda i eit punkt i det eine kamera, og i eit anna punkt i det andre. Sidan desse punkta svarar til det same punktet, vil dei vere korresponderande punkt. Det er denne korrespondansen me vil utforske nærmare. Me vil også finne ut kor mange korresponderande punkt ein må ha for å finne ei endeleg løysing for den essensielle matrisa, og dermed kunne finne kameraforskyvinga. Deretter vil me finne ut kor mange moglegheitar dette gir, generelt. For å finne svaret på dette, må me introdusere fundamentalmatrisa og den essensielle matrisa. Me byrjar med å gi ein introduksjon til det projektive planet, og deretter ser me på projektiv geometri og transformasjonar i tre dimensjonar. Dette brukar me når me skal sjå på ulike kameramodellar, og den essensielle matrisa. For å kunne dekomponere den essensielle matrisa og finne kameraforskyvinga, må me løyse ei mengde med polynomlikningar. For å kunne finne ei løysing på denne mengda med polynomlikningar, i fleire variablar, må ein bruke gröbnerbasar.
Wed, 01 Jun 2016 00:00:00 GMThttp://hdl.handle.net/1956/126582016-06-01T00:00:00ZMisoppfatningar i algebra hjå elevar på første trinn i vidaregåande skulehttp://hdl.handle.net/1956/12652
Misoppfatningar i algebra hjå elevar på første trinn i vidaregåande skule
Audestad, Torbjørn
Master thesis
Denne oppgåva handlar om misoppfatningar i
algebra hjå 1.års-elevar i vidaregåande skule.
Målet med oppgåva har vore å gje meg som
matematikklærar større innsikt i
misoppfatningar og vanskar elevar møter i
algebra. Eg har prøvd setja meg inn i relevant
teori på førehand og undervegs i arbeidet, og
eg har samanlikna resultata eg har funne med
andre tidlegare arbeid. Det var totalt 91
elevar som deltok i denne studien. Elevane
fulgte matematikkkursa 1P-Y, 1P eller 1T,
hovudsakleg ved den skulen eg jobbar på.
Masterprosjektet følgjer eit mixed methods-
design. Den kvantitative delen består av ei
kartleggingsundersøking der elevane sine
dugleikar i algebra vart kartlagt gjennom ei
rekkje oppgåver. Resultata vart registrerte i
eit Excel-dokument. På bakgrunn av desse dataa
vart det utarbeidd ein intervjuguide som vart
brukt i intervju med eit utval på ti av elevane
som deltok i studien. Eg intervjua elevar for å
få ei djupare innsikt i nokre av
misoppfatningane som kom fram i testen.
Intervjua utgjorde den kvalitative delen av
studien, og gav eit meir nyansert innsyn i
elevane si tenking.
Resultata av kartleggingsprøva viste tydelege
teikn på at det rådde ein del misoppfatningar
hjå elevane. Mange av elevane sat utelukkande
med prosedyrekunnskapar, dei viste lita eller
ingen forståing for kva dei gjorde. Dei elevane
som som såg ut til å ha manglande strukturell
kunnskap, vart avslørte i oppgåver der
forståing av variablar var ein føresetnad for å
få til oppgåvene. Det viste seg å vera ein
samanheng mellom grunnleggjande matematiske
dugleikar og dei vanskane elevane møter i
algebra. Hol i kunnskapen i aritmetikk gjer
vegen vidare med algebra alt anna en enkel.
Eg oppdaga at det var fleire sider ved
misopfatningane eg studerte som eg ikkje hadde
sett for meg på førehand. Elevane hadde ulike
forklaringar på det eg trudde skuldast ei type
misoppfatning. For meg var derfor den rikaste
erfaringa i dette arbeidet kor komplisert det
er å finna ut av kva eleven faktisk tenkjer når
han løyser oppgåver i matematikk. Er det ei
misoppfatning som faktisk ligg bak eit
feilsvar, eller er det feil som skuldast slurv,
feil knytt til System 1-tankegang, eller er det
rett og slett kognitiv svikt? Dette viste seg
altså vera samansatt å finna ut av.
Det viser seg av dataa mine at ei oppgåvene i
algebra som vert oppfatta som vanskeleg for ein
1P-Y-elev òg vil vera det for ein 1P- eller 1T-
elev, men då i ulik grad. Misoppfatningar er
med andre ord kompliserte, men dess viktigare å
ha kjennskap til for å få til ei best mogleg
tilpassa undervisning for eleven.
Thu, 02 Jun 2016 00:00:00 GMThttp://hdl.handle.net/1956/126522016-06-02T00:00:00ZStochastic unit-root processeshttp://hdl.handle.net/1956/12650
Stochastic unit-root processes
Dyrhovden, Sigve Brix
Master thesis
In practice there are many types of processes that have near unit roots and are very difficult to distinguish from standard unit-root processes, given a finite sample. In our thesis we study a particular class of such processes, also called STUR processes. They have a root that is stochastic and varying around unity. We also study standard unit-root processes and cointegration. In cointegration analysis, the first step is to test whether the variables of interest have a unit root, and thus, unit root tests are used in application of cointegration.
Wed, 01 Jun 2016 00:00:00 GMThttp://hdl.handle.net/1956/126502016-06-01T00:00:00ZSupervised Continuous Max-flow for Segmentation of Images with Intensity Inhomogeneitieshttp://hdl.handle.net/1956/12643
Supervised Continuous Max-flow for Segmentation of Images with Intensity Inhomogeneities
Guo, Junjie
Master thesis
In this thesis we propose a regional adaptive active
contour model for segmenting images with intensity
inhomogeneities effectively. This model includes a
regularization term, a global energy term and a local
energy term. The regularization term is used to control
the length of the border of the characteristic function.
At the early stage, the global term which is larger than
the local term can give us a good initial segmentation of
the image quickly. In the later phase of the process, the
local energy term dominating the whole function energy
can localize the precise position of the object of
interest in image with intensity inhomogeneities. An
algorithm based on supervised continuous max-flow method
is developed to solve this problem robustly and validly.
A large number of experiments are presented to show how
this proposed method behaves on different images.
Thu, 26 May 2016 00:00:00 GMThttp://hdl.handle.net/1956/126432016-05-26T00:00:00ZHigher Hochschild homology is not a stable invarianthttp://hdl.handle.net/1956/12630
Higher Hochschild homology is not a stable invariant
Tenti, Andrea
Master thesis
The purpose of this thesis is to study the higher
Hochschild homology of some rational algebras. We
know that for some algebras the higher Hochschild
homology is a stable invariant. Studying the
homology over the spheres and over the torus, we
can deduce that this is not true in the most
general sense. We will give a counterexample
based on the algebra of the dual numbers over a
field of characteristic 0. Moreover we will study
the equivariant structure of the iterated
Hochschild homology for some particular algebras
as a toy model, in order to shed some lights on
the limits of the topological version of the
Hochschild homology, which plays a key role in
the understanding of the chromatic shift of K-
theory.
Fri, 27 May 2016 00:00:00 GMThttp://hdl.handle.net/1956/126302016-05-27T00:00:00ZVurdering for læring i matematikk - vurderingspraksis i 1YT i to klassar på elektrofaghttp://hdl.handle.net/1956/12628
Vurdering for læring i matematikk - vurderingspraksis i 1YT i to klassar på elektrofag
Årdal, Jill Berit
Master thesis
Denne oppgåva handlar om formativ vurdering i matematikk. I problemstillinga spør ein etter i kva grad kapittel- og lekseprøver i 1YT har ein formativ funksjon for elevar på elektrofag. Ein skil mellom formål og funksjon med utgangspunkt i at vurderingssituasjonar ikkje alltid fungerer etter formålet. Studien tek utgangspunkt i kvalitativ metode og byggjer på vurderingspraksis i to eigne klassar. Datagrunnlaget er i hovudsak elevar sine eigenvurderingar etter kapittelprøver og opptak av fagsamtalar.
Tue, 31 May 2016 00:00:00 GMThttp://hdl.handle.net/1956/126282016-05-31T00:00:00ZAlternative læringsmåter for derivasjon i R1 - en komparativ analysehttp://hdl.handle.net/1956/12626
Alternative læringsmåter for derivasjon i R1 - en komparativ analyse
Aksdal, Tom
Master thesis
I denne masteroppgaven har jeg studert elever som har arbeidet med derivasjon i matematikk R1. Formålet har vært å sammenligne to alternative læringsmåter med tradisjonell oppgaveregning, med tanke på læringsutbytte på kort sikt og på litt lengre sikt. De alternative læringsmåtene var aktiv memorering og problemløsning. Alle elevene som deltok hadde hatt en relativt tradisjonell opplæring i derivasjon på forhånd av sine respektive faglærere.
Motivasjonen min for å prøve ut disse alternative læringsmåtene var erfaringer fra R2, der mye av kunnskapen om derivasjon fra R1 tilsynelatende var helt glemt hos svært mange av R2-elevene mine i løpet av sommerferien. Og selv etter repetisjon var det fortsatt ganske mange som ikke mestret derivasjon slik man forventer i R2.
Det ble gjennomført et derivasjonsprosjekt, der 133 elever fordelt på sju R1-klasser i Bergensregionen deltok. Hver klasse ble delt inn i tre grupper, og hver gruppe arbeidet med hver sin faste læringsmåte over en periode på tre dobbelttimer. I denne perioden ble det gjennomført skriftlige tester. Etter prosjektet ble det gjennomført en spørreundersøkelse, og utvalgte elever ble intervjuet. Etter ca. fire måneder ble det gjennomført en skriftlig posttest. Data fra de 76 elevene som hadde deltatt på alt dette ble deretter analysert og sammenlignet.
Elevene som var med på derivasjonsprosjektet så gjennomgående ut til å oppfatte derivasjon som vanskelig. Det virket dessuten som at de først og fremst forbandt derivasjon med regelbruk og prosedyrer for å finne ekstremalpunkt.
Jentene som jobbet med alternative læringsmåter så ut til å ha en bedre utvikling på testene enn guttene. Likevel var det en overvekt av gutter som hadde et positivt syn på å jobbe med slike alternative læringsmåter, og de følte i større grad enn jentene at de hadde hatt et positivt læringsutbytte av å være med på derivasjonsprosjektet.
Selv om det var knyttet en viss usikkerhet til resultatene, ser det ut som om aktiv memorering kan gi økt læringsutbytte på kort sikt, mens problemløsning kan se ut til å gi økt læringsutbytte på litt lengre sikt. Det kan altså virke som at matematikklærere som underviser tradisjonelt om derivasjon i R1 med fordel kan vurdere å supplere denne opplæringen med en av de alternative metodene som er testet ut i denne studien.
Wed, 01 Jun 2016 00:00:00 GMThttp://hdl.handle.net/1956/126262016-06-01T00:00:00ZElevers holdninger til matematikk i videregående skolehttp://hdl.handle.net/1956/12598
Elevers holdninger til matematikk i videregående skole
Ludvigsen, Gerd Anne
Master thesis
Målet med oppgaven har vært å få en dypere innsikt og mer kunnskap om elevers holdninger til matematikken på et høyere nivå i videregående skole. Ut fra et sosialkonstruktivistisk perspektiv har fokus vært rettet mot elevenes utvikling av egen holdning til matematikken ut fra deres erfaringer med matematikken og matematikkrelaterte følelser for faget. Seks elever på Vg3-nivå i matematikk på realfag i tilknytning til programfaget R2 på utdanningsprogram for studiespesialisering har deltatt i undersøkelsen. Metodevalget er kvalitativt. Datainnsamlingen er gjennomført i form av semistrukturerte intervju. Intervjuene er transkribert og kategorisert. Det transkriberte kategoriserte datamaterialet har dannet grunnlaget for videre analyse og tolkning. Under analysen og tolkningen har jeg «søkt» etter nøkkelfaktorer som kan ha betydning for elevers utvikling av egen holdning til matematikken. Det er tatt utgangspunkt i den utviklede tredimensjonale holdningsmodellen som bygger på forskningsarbeid på holdninger til matematikken av Di Martino Pietro og Zan Rosetta (2010; 2011). Modellen er inndelt i elevens matematikkrealterte følelser, elevens syn på matematikk og elevens oppfatning av kompetanse. Modellen vektlegger sammenhengen mellom: følelser og syn på matematikken, følelser og oppfattet kompetanse og sammenhengen mellom syn på matematikken og oppfattet kompetanse. Funnene viser at elevene i hovedsak har en positiv holdning til matematikken, og de fleste elevene har utviklet økt interesse for matematikkfaget. Det framkommer at interessen er situasjonell og individuell, og situasjonell interesse ser ut til å ha endret seg hos enkelte til en holdningsbasert interesse. Videre ser det ut som at faglig selvoppfatning og ulike oppfatninger om hva som er viktig for å lykkes i matematikk kan ha betydning for elevenes utvikling av egen holdning. Nøkkelfaktorer som er framtredende, og som kan ha betydning for elevens utvikling av egen holdning viser seg å være: a) Positive matematikkrelaterte følelser og instrumentelt syn på matematikk b) Elevene har utviklet egen holdning til matematikken ut fra økt interesse for faget c) Matematikkrelaterte følelser og oppfatning av egen kompetanse d) Fordypning i matematikk på grunn av framtidsplaner for videre studier og yrke.
Wed, 01 Jun 2016 00:00:00 GMThttp://hdl.handle.net/1956/125982016-06-01T00:00:00ZAnalysis of non-linear shallow water equations discretized on Arakawa C-gridshttp://hdl.handle.net/1956/12580
Analysis of non-linear shallow water equations discretized on Arakawa C-grids
Eidsvåg, Espen Knudsen
Master thesis
To model the behaviour of flow and waves in
three dimensions, one may use the Navier-Stokes
equations. These equations leads to the shallow
water equations, when the depth is much smaller
than the wavelength. The shallow water
equations are depth-integrated Navier-Stokes
equations, and they depict in general a two
dimensional wave which can not vary with the
depth.
Storm surges and tidal waves are described by
the shallow water equations. When we want to
make a forecast of these waves, it is
beneficial to use shallow water equations
discretized on a Arakawa C-grid. Fourier
methods have been used to study the stability
of the solutions for constant depths. In this
master thesis, we will study the stability of
numerical methods for solving the shallow water
equations, investigating the eigenvalues of the
propagation matrices of a few grid cells. The
effects of variable depths and non-linear terms
are addressed. For the constant depths cases,
the eigenvalues are purely imaginary. In the
non-linear cases, the eigenvalues may have non-
zero real components, which may give growth or
damping of the solutions. By building on the
two-cell case, the three- and the four-cell
cases are explored in detail. With the three-
cell case we introduce the non-linear term,
which have the potential to change the
stability.
For this case, we show a skew-symmetric form
for the non-linear shallow water equations by
combining the flux and advective forms in the
three cell case. This form does not work on
variable depths. The four-cell case is
discussed to great extent, as it is set up
differently than the two- and three-cell case.
The behaviour of the eigenvalues have been
important to study the stability of the cases.
To understand the modes of the cases, the
eigenvectors have been interpreted.
The condition of skew-symmetry for the non-
linear variable depths cases might be a too
strong restriction, as the modes do not
disperse to different frequencies. This is
contrary to the behaviour of real waves, where
we have interaction between different modes.
Wed, 01 Jun 2016 00:00:00 GMThttp://hdl.handle.net/1956/125802016-06-01T00:00:00ZIs there a U-shaped relationship between physical activity in leisure time and risk of chronic low back pain? A follow-up in the HUNT Studyhttp://hdl.handle.net/1956/12545
Is there a U-shaped relationship between physical activity in leisure time and risk of chronic low back pain? A follow-up in the HUNT Study
Heuch, Ingrid; Heuch, Ivar; Hagen, Knut; Zwart, John-Anker
Journal article
<p>Background: Physical activity in leisure time is often considered to have favourable effects on the risk of low back pain (LBP), but demonstrating a definite association in epidemiological studies has proven difficult. The purpose of the present study was to explore associations between physical activity and risk of chronic LBP in an adult population and to investigate whether relationships are limited to certain age groups or to females or males. A particular objective was to determine whether support could be found for a U-shaped relationship, with both low and high activity levels carrying greater risk.</p>
<p>Methods: The relationship between physical activity and risk of chronic LBP was examined in a Norwegian prospective study using data from the community-based HUNT2 and HUNT3 surveys. Participants were 9616 women and 8452 men without LBP at baseline, who reported after 11 years whether they suffered from LBP. Associations between baseline physical activity in leisure time and risk of chronic LBP at end of follow-up were evaluated by generalized linear modelling with adjustment for potential confounders.</p>
<p>Results: Significant associations between leisure time physical activity and risk were observed in both sexes after age adjustment, mainly suggesting inverse relationships. Women participating in hard physical activity 1&ndash;2 h per week had a relative risk (RR) of chronic LBP of 0.81 (95 % CI 0.71&ndash;0.93) compared to those with only light physical activity less than 1 h per week. The corresponding RR in men was 0.71 (95 % CI 0.60&ndash;0.85). After adjustment for education, employment, occupational activity, body mass index (BMI) and smoking, significant relationships could only be demonstrated in those aged 50 years or more at baseline. The associations differed between female educational groups, with more U-shaped relationships being observed among women with basic education only.</p>
<p>Conclusion: No strong support was found overall for U-shaped relationships. However, no further general decrease in risk was seen among those with 3 h or more of hard physical activity per week. The contrasts observed between female educational groups may reflect different preferences regarding specific strenuous activities. Men aged 50 years or more seem in particular to benefit from hard physical activities.</p>
Mon, 11 Apr 2016 00:00:00 GMThttp://hdl.handle.net/1956/125452016-04-11T00:00:00ZMathematical Modeling and Simulation for Microbial Enhanced Oil Recovery Optimizationhttp://hdl.handle.net/1956/12544
Mathematical Modeling and Simulation for Microbial Enhanced Oil Recovery Optimization
Kassa, Abay Molla
Master thesis
In this study, we developed a mathematical model for 2D microbial EOR technology. It consists of an average pressure formulation of two-phase flow and component advection-diffusion reactive transport models. An IFT reduction mechanism has been used to couple porous media flow and transport phenomenon. A cell centered control volume and backward Euler methods were applied to discretize the flow and transport equations. An iterative linearization technique is implemented in Matlab to solve the resulting system of non-linear algebraic equations. Numerical experiments were conducted on two-phase flow, advection-diffusion transport and the coupled microbial EOR models for different mobility and capillary pressure parameterizations. Moreover, we have proposed a new microbial EOR optimization model to maximize the sweep efficiency by controlling the slug size injection strategy. In this thesis, we have also proposed a new optimization algorithm based on gradient accent and iterative linearization methods to handle the microbial EOR optimization model numerically. A number of illustrative numerical experiments have been done to analyse the effect of slug size and cost of microbial concentration.
Wed, 01 Jun 2016 00:00:00 GMThttp://hdl.handle.net/1956/125442016-06-01T00:00:00ZPost-lie algebras and isospectral flowshttp://hdl.handle.net/1956/12467
Post-lie algebras and isospectral flows
Ebrahimi-Fard, Kurusch; Lundervold, Alexander; Mencattini, Igor; Munthe-Kaas, Hans
Journal article
In this paper we explore the Lie enveloping algebra of a post-Lie algebra derived from a classical R-matrix. An explicit exponential solution of the corresponding Lie bracket flow is presented. It is based on the solution of a post-Lie Magnus-type differential equation.
Fri, 20 Nov 2015 00:00:00 GMThttp://hdl.handle.net/1956/124672015-11-20T00:00:00ZSolute transport in aquifers with evolving scale heterogeneityhttp://hdl.handle.net/1956/12449
Solute transport in aquifers with evolving scale heterogeneity
Suciu, Nicolae; Attinger, Sabine; Radu, Adrian Florin; Vamo̧s, Cǎlin; Vanderborght, Jan; Vereecken, Harry; Knabner, Peter
Journal article
Transport processes in groundwater systems with spatially heterogeneous properties often exhibit anomalous behavior. Using first-order approximations in velocity fluctuations we show that anomalous superdiffusive behavior may result if velocity fields are modeled as superpositions of random space functions with correlation structures consisting of linear combinations of short-range correlations. In particular, this corresponds to the superposition of independent random velocity fields with increasing integral scales proposed as model for evolving scale heterogeneity of natural porous media [Gelhar, L. W. Water Resour. Res. 22 (1986), 135S-145S]. Monte Carlo simulations of transport in such multi-scale fields support the theoretical results and demonstrate the approach to superdiffusive behavior as the number of superposed scales increases.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/1956/124492015-01-01T00:00:00ZConsistency issues in pdf methodshttp://hdl.handle.net/1956/12420
Consistency issues in pdf methods
Suciu, Nicolae; Schüler, Lennart; Radu, Adrian Florin; Attinger, Sabine; Vamo̧s, Cǎlin; Knabner, Peter
Journal article
Concentrations of chemical species transported in random environments need to be statistically characterized by probability density functions (PDF). Solutions to evolution equations for the one-point one-time PDF are usually based on systems of computational particles described by It^o equations. We establish consistency conditions relating the concentration statistics to that of the It^o process and the solution of its associated Fokker-Planck equation to that of the PDF equation. In this frame, we use a recently proposed numerical method which approximates PDFs by particle densities obtained with a global random walk (GRW) algorithm. The GRW-PDF approach is illustrated for a problem of contaminant transport in groundwater.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/1956/124202015-01-01T00:00:00ZModeling and Simulation of Microbial Enhanced Oil Recovery: A New Approach which Includes the Role of Interfacial Areahttp://hdl.handle.net/1956/12297
Modeling and Simulation of Microbial Enhanced Oil Recovery: A New Approach which Includes the Role of Interfacial Area
Landa Marban, David
Master thesis
<p>In today's world, oil remains the main source of energy. After discovering a petroleum reservoir, one
can extract about 30% of the oil by using and maintaining the initial pressure in the reservoir through
water ooding (first and second phase oil recovery). Nevertheless, 60-70% of oil remains in the reservoir
after this, so called conventional recovery, so enhanced oil recovery (EOR) is strongly needed. Microbial
enhanced oil recovery (MEOR) is an environmentally friendly and very promising EOR technology.</p>
<p>In this thesis we implement a non-standard model for MEOR that includes the interfacial area between
the oil and water. We consider the two-phase ow equations and multicomponent reactive transport
equations for three components: bacteria, nutrients and biosurfactants. The growth of bacteria affects the
properties of the medium (porosity, permeability), up to the extreme case of pore clogging (bioclogging)
and generation of new paths. Biosurfactants lower the interfacial tension, which improves the oil recovery.
We consider in this thesis, for the fi
The motivation to include the interfacial area in the model is to eliminate the hysteresis in the capillary
pressure relationship and model that biosurfactants are mainly living at the oil-water interface.</p>
<p>A typical mathematical model for MEOR consists on nonlinear coupled partial differential equations
(PDEs) and ordinary differential equations (ODEs). The spatial discretization is obtained using finite
differences (FD) and two-point ux approximation (TPFA), and the time discretization using backward
Euler (BE). We make the implementation for 1-D and 2-D domains using cell-centered grids. We present
in detail the discretization of the equations, including the treatment of the boundary conditions. After
the discretization in space and time, the problems to be solved at each time step are still nonlinear. For
solving these equations, we use an implicit scheme that considers a linear approximation of the capillary
pressure gradient, which results in an efficient and stable scheme. The code for running the simulations is
written in MATLAB. Following this, we test the code with analytic solutions and benchmark simulations.</p>
<p>We consider a 1-D porous medium where we study the spatial distribution and the evolution in time of the
average pressure, water saturation, oil-water interfacial area, capillary pressure, porosity, permeability
ratio, residual oil water saturation and bacterial, nutrient and biosurfactant concentrations. After, we
make a sensitivity analysis in order to examine the effects of the relevant model parameters. Finally, we
make simulations considering a porous medium with a thief zone, which is a 2-D problem.</p>
<p>A comprehensive model, including two-phase ow, bacteria, nutrients and biosurfactants was implemented.
The model has been tested using analytic and benchmark problems. For the first time, the
role of interfacial area in MEOR was studied. We showed that different predictions of oil recovery are
obtained by including the availability of interfacial area in the model. Nevertheless, it is necessary to do
more experiments in the laboratory in order to compare with the numerical simulations and validate the
model assumptions.</p>
Tue, 31 May 2016 00:00:00 GMThttp://hdl.handle.net/1956/122972016-05-31T00:00:00ZMeningsfuld begynderundervisning i matematisk analysehttp://hdl.handle.net/1956/12204
Meningsfuld begynderundervisning i matematisk analyse
Andresen, Mette Susanne
Journal article
This article presents a study of a minor reversion of a first year course in
calculus at Mathematics Department, University of Bergen, Norway. The
lecturer’s goal was to force the students to take responsibility for their own
learning. The article brings reflections upon the lecturer’s professional
autonomy and background for the revision of the course. The study was neither
quantitative nor comparative. Data do not suggest that the students’ working
habits had changed. The rate of passing students, though, had increased.; Artiklen præsenterer studiet af en mindre ændring i organiseringen af undervisningen på et førsteårskursus, gennemført i efteråret 2013 på Matematisk Institut, UiB (Universitetet i Bergen, Norge). Underviseren ønskede at omorganisere kurset, så de studerende blev mere ansvarlige for deres egen læring. Artiklen bringer didaktiske refleksioner over underviserens belæg for den gennemførte omorganisering. Studiet, som præsenteres, var hverken kvantitativt eller komparativt. Det indsamlede datamateriale tyder ikke på, at de studerendes arbejdsformer eller indbyrdes samarbejde adskilte sig fra tidligere år. Men de studerende på kurset fik bedre resultater til eksamen end tidligere år, idet gennemførelsesprocenten var højere.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/1956/122042015-01-01T00:00:00ZGeneral Slit Stochastic Löwner Evolution and Conformal Field Theoryhttp://hdl.handle.net/1956/11975
General Slit Stochastic Löwner Evolution and Conformal Field Theory
Tochin, Alexey
Doctoral thesis
Fri, 22 Jan 2016 00:00:00 GMThttp://hdl.handle.net/1956/119752016-01-22T00:00:00ZConvergence of a Cell-Centered Finite Volume Discretization for Linear Elasticityhttp://hdl.handle.net/1956/11884
Convergence of a Cell-Centered Finite Volume Discretization for Linear Elasticity
Nordbotten, Jan Martin
Journal article
We show convergence of a cell-centered finite volume discretization for linear elasticity. The discretization, termed the MPSA method, was recently proposed in the context of geological applications, where cell-centered variables are often preferred. Our analysis utilizes a hybrid variational formulation, which has previously been used to analyze finite volume discretizations for the scalar diffusion equation. The current analysis deviates significantly from the previous in three respects. First, additional stabilization leads to a more complex saddle-point problem. Second, a discrete Korn's inequality has to be established for the global discretization. Finally, robustness with respect to the Poisson ratio is analyzed. The stability and convergence results presented herein provide the first rigorous justification of the applicability of cell-centered finite volume methods to problems in linear elasticity.
Thu, 19 Nov 2015 00:00:00 GMThttp://hdl.handle.net/1956/118842015-11-19T00:00:00ZTwo-phase flow in porous media: dynamic capillarity and heterogeneous mediahttp://hdl.handle.net/1956/11835
Two-phase flow in porous media: dynamic capillarity and heterogeneous media
van Duijn, Cornelius J.; Cao, Xiulei; Pop, Iuliu Sorin
Journal article
We investigate a two-phase porous media flow model, in which dynamic effects are taken into account in phase pressure difference. We consider a one-dimensional heterogeneous case, with two adjacent homogeneous blocks separated by an interface. The absolute permeability is assumed constant, but different in each block. This may lead to the entrapment of the non-wetting phase (say, oil) when flowing from the coarse material into the fine material. We derive the interface conditions coupling the models in each homogeneous block. In doing so, the interface is approximated by a thin porous layer, and its thickness is then passed to zero. Such results have been obtained earlier for standard models, based on equilibrium relationship between the capillary pressure and the saturation. Then, oil is trapped until its saturation on the coarse material side of the interface exceeds an entry value. In the non-equilibrium case, the situation is different. Due to the dynamic effects, oil may still flow into the fine material even after the saturation drops under the entry point, and this flow may continue for a certain amount of time that is proportional to the non-equilibrium effects. This suggests that operating in a dynamic regime reduces the account of oil trapped at interfaces, leading to an enhanced oil recovery. Finally, we present some numerical results supporting the theoretical findings.
Tue, 11 Aug 2015 00:00:00 GMThttp://hdl.handle.net/1956/118352015-08-11T00:00:00ZAn Open-Source Toolchain for Simulation and Optimization of Aquifer-Wide CO2 Storagehttp://hdl.handle.net/1956/11782
An Open-Source Toolchain for Simulation and Optimization of Aquifer-Wide CO2 Storage
Andersen, Odd; Lie, Knut-Andreas; Nilsen, Halvor Møll
Journal article
Planning and execution of large-scale, aquifer-wide CO2 storage operations will require extensive use of computer modelling and simulations. The required computational tools will vary depending on the physical characteristics of the targeted aquifer, the stage of the project, and the questions asked, which may not always be anticipated in advance. In this paper, we argue that a one-size-fits-all simulation tool for the modelling CO2 storage does not exist. Instead, we propose an integrated toolchain of computational methods that can be used in a flexible way to set up adaptive workflows. Although a complete toolchain will require computational methods at all levels of complexity, we further argue that lightweight methods play a particularly important role in addressing many of the relevant questions. We have implemented a number of such simplified methods in MRST-co2lab, a separate module to the open-source MATLAB Reservoir Simulation Toolbox (MRST). In particular, MRST-co2lab contains percolation-type methods that within seconds can identify structural traps and their catchment areas and compute spill paths and rough estimates of trapping capacity. The software also offers two-phase simulators based on vertical-equilibrium assumptions that can forecast structural, residual, and solubility trapping in a thousand-year perspective and are orders of magnitude faster than traditional 3D simulators. Herein, we apply these methods on realistic, large-scale datasets to demonstrate their capabilities and show how they can be used in combination to address optimal use of open aquifers for large-scale storage.
Thu, 04 Feb 2016 00:00:00 GMThttp://hdl.handle.net/1956/117822016-02-04T00:00:00ZUpscaling of Nonisothermal Reactive Porous Media Flow under Dominant Péclet Number: The Effect of Changing Porosityhttp://hdl.handle.net/1956/11768
Upscaling of Nonisothermal Reactive Porous Media Flow under Dominant Péclet Number: The Effect of Changing Porosity
Bringedal, Carina; Berre, Inga; Pop, Iulio Sorin; Radu, Florin A.
Journal article
Motivated by rock-fluid interactions occurring in a geothermal reservoir, we present a two-dimensional pore scale model of a thin strip consisting of void space and grains, with fluid flow through the void space. Ions in the fluid are allowed to precipitate onto the grains, while minerals in the grains are allowed to dissolve into the fluid, taking into account the possible change in the aperture of the strip that these two processes cause. Temperature variations and possible effects of the temperature in both fluid density and viscosity and in the mineral precipitation and dissolution reactions are included. For the pore scale model equations, we investigate the limit as the width of the strip approaches zero, deriving one-dimensional effective equations. We assume that the convection is dominating over diffusion in the system, resulting in Taylor dispersion in the upscaled equations and a Forchheimer-type term in Darcy's law. Some numerical results where we compare the upscaled model with three simpler versions are presented: two still honoring the changing aperture of the strip but not including Taylor dispersion, and one where the aperture of the strip is fixed but contains dispersive terms.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/1956/117682016-01-01T00:00:00ZOppfatningar og misoppfatningar i sannsyn: Ein mixed methods-studie av elevar på vidaregåande trinn – yrkesfagleg retninghttp://hdl.handle.net/1956/11748
Oppfatningar og misoppfatningar i sannsyn: Ein mixed methods-studie av elevar på vidaregåande trinn – yrkesfagleg retning
Handegård, Tone
Master thesis
Denne masteroppgåva handlar om korleis elevar på vidaregåande trinn – yrkesfagleg retning forstår omgrep innafor sannsyn. Føremålet har vore å avdekkja kva oppfatningar og misoppfatningar elevane har og kva løysingsstrategiar elevane nyttar når dei svarar på spørsmåla. Vidare har eg undersøkt i kva grad funna mine samsvarar med tidlegare funn som er referert i litteraturen i tillegg til å sjå nærare på korrelasjon mellom karakter og resultat.
I studien er det nytta mixed methods for datainnsamlinga. Datainnsamlinga vart gjennomført i januar og februar 2015. Det vart først utført ei kvantitativ spørjeundersøking der 93 elevar deltok. Om lag 2/3 av desse går på vg1 i ulike yrkesfaglege retningar medan om lag 1/3 går på påbygg til generell studiekompetanse. Etterpå vart det utført ein kvalitativ studie der 9 elevar vart intervjua. Ingen av elevane hadde gjennomgått opplæring i sannsyn rett før datainnsamlinga. Spørsmåla byggjer i stor grad på spørsmål frå tidlegare undersøkingar slik at dei til ei viss grad kan seiast å vera kvalitetssikra.
Alle misoppfatningane som eg ynskte å testa, synte seg å vera til stades i varierande grad hjå elevane. Dei talmessige resultata frå undersøkinga mi viser elles at det i stor grad er samanfallande med det forskarar har funne tidlegare, trass i at undersøkingane har stor variasjon i alder, skulebakgrunn og verdsdeler.
Elevane nytta ulike løysingsstrategiar som vi og finn att i litteraturen og tidlegare forsking. Dette gjeld særleg bruk utfallstilnærming og 50-50 tilnærming. Gjennom intervjua fekk eg og avdekka at elevane gjerne stolar på intuisjonen sin og det er først når dei må tenkja i gjennom kva som står i spørsmålet, at dei kan gje eit normativt riktig svar. Det er særleg innafor sannsyn at bruk av intuisjon leier oss inn i ei misoppfatning. Vi nyttar som oftast det vi har lært tidlegare.
Fri, 01 Jan 2016 00:00:00 GMThttp://hdl.handle.net/1956/117482016-01-01T00:00:00ZTwo-scale preconditioning for two-phase nonlinear flows in porous mediahttp://hdl.handle.net/1956/11747
Two-scale preconditioning for two-phase nonlinear flows in porous media
Skogestad, Jan Ole; Keilegavlen, Eirik; Nordbotten, Jan Martin
Journal article
Solving realistic problems related to flow in porous media to desired accuracy may be prohibitively expensive with available computing resources. Multiscale effects and nonlinearities in the governing equations are among the most important contributors to this situation. Hence, developing methods that handle these features better is essential in order to be able to solve the problems more efficiently. Focus has until recently largely been on preconditioners for linearized problems. This article proposes a two-scale nonlinear preconditioning technique for flow problems in porous media that allows for incorporating physical intuition directly in the preconditioner. By assuming a certain dominant physical process, this technique will resemble upscaling in the equilibrium limit, with the computational benefits that follow. In this study, the method is established as a preconditioner with good scalability properties for challenging problems regardless of dominant physics, thus laying the foundation for further studies with physical information in the preconditioner.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/1956/117472015-01-01T00:00:00ZTrigonometric interpolation on lattice gridshttp://hdl.handle.net/1956/11732
Trigonometric interpolation on lattice grids
Sørevik, Tor; Nome, Morten
Journal article
In this paper we construct non-aliasing interpolation spaces and Lagrange functions for lattice grids. We argue that lattice grids are good for trigonometric interpolation and support this claim by numerical experiments. A greedy algorithm allows us to embed hyperbolic crosses in our interpolation spaces, and numerical experiments indicate that lattice grids are at least as good as sparse grids for trigonometric interpolation. A straightforward FFT-algorithm for functions sampled on lattice grids allows for fast computation and good approximation.
Thu, 01 Jan 2015 00:00:00 GMThttp://hdl.handle.net/1956/117322015-01-01T00:00:00ZWave breaking in long wave models and undular boreshttp://hdl.handle.net/1956/11668
Wave breaking in long wave models and undular bores
Brun, Mats Kirkesæther
Master thesis
Bores are a well known phenomena in fluid mechanics, although
their occurrence in nature is relatively rare. The circumstances in
which they occur is usually when a tidal swell causes a difference
in surface elevation in the mouth of a river, or narrow bay, causing
long waves to propagate upstream. The term 'tidal bore' is also
frequently used in this context. Depending on the conditions the
bore may take on various forms, ranging from a smooth wavefront
followed by a smaller wave train, to one singe breaking wavefront.
Some noteworthy locations where tidal bores can be found include
the River Seine in France, the Petitcodiac River in Canada, and the
Qiantang River in China. Common for all these locations is a large
tidal range. Bores, when powerful enough, can produce particularly
unsafe environments for shipping, but at the same time popular
opportunities for river surfing.
As found by Favre in 1935 by wave tank experiments, the strength
of the bore can be determined by the ratio of the incident water
level above the undisturbed water depth to the undisturbed water
depth. Denoting this ratio by $\alpha$, bores can occur in one of
three categories: If $\alpha$ is less than 0.28 the bore is purely
undular, and will feature oscillations downstream of the bore front.
If $\alpha$ is between 0.28 and 0.75 the bore will continue to
feature oscillations, but one or more waves behind the bore front
will start to break. If $\alpha$ is greater than 0.75 the bore is
completely turbulent, and can no longer be described by the
standard potential flow theory.
The goal of this report is to simulate the time evolution of an
undular bore through numerical experiments, using a dispersive
nonlinear shallow water theory, in particular the Korteweg-deVries
(KdV) equation. This is a third order nonlinear partial differential
equation, where the dependent variable describes the
displacement of the free surface. When deriving this equation, an
expression for the velocity field of the flow is also available. This
can be calculated at any point in the fluid, as long as the
displacement of the surface is known. Thus, solving the KdV
equation also yields the fluid particle velocity field, which can be
used to calculate fluid particle trajectories, as done by Bjørkvåg
and Kalisch, but also to formulate a breaking criterion. By applying
this breaking criterion to the undular bore, the onset of breaking,
and thus also a maximum allowable wave height (due to the
nonlinearity of the model equation), can be computed numerically.
This criterion can also be applied to the exact traveling wave
solutions of the KdV equation, namely the 'solitary wave' solution
and the 'cnoidal wave' solution. These are waves of constant shape
traveling at constant velocity, thus applying the breaking criterion
yields a maximum height for which they can exist.
The theory leading to the formulation of the KdV equation is also
included, in addition to formulation of the linearized and shallow
water equations. These, however, serve only as 'stepping stones'
towards the higher order Boussinesq equations, and are not used
in any further calculations.
Fri, 18 Dec 2015 00:00:00 GMThttp://hdl.handle.net/1956/116682015-12-18T00:00:00ZSubsea pump vibration analysis for predictive maintenancehttp://hdl.handle.net/1956/11666
Subsea pump vibration analysis for predictive maintenance
Rouzbahaneh, Mahdi
Master thesis
The goal of this study is to compare vibration signals of subsea pumps from two different time periods. These signals are gathered for predictive maintenance reasons. Vibration analysis is one of the most efficient tools for machine diagnostics. Vibration analysis has been an active area of research in the past few decades. In this thesis an attempt has been made to cover all the common vibration analysis methods to find more reliable results. Since a pump includes many rotating components, presence of a kind of periodicity is expected in the vibration signal. Local defects in each cycle of rotation cause an impulse in the vibration signal. Therefore high impulsivity in the vibration signal is an indicator of fault. This Impulsivity is not clearly observed in the vibration signal. The impulse response function of the transmission path deforms an impulse in the source into a spread segment in the vibration signal. Furthermore the impulses are more hidden when the period of impulses is shorter than the width of the impulse response function. Detection of impluses is more subtle due to the presence of background noise. Our methods mainly focus on these two obstacles to achieve impulsivity of source signal. These Impulses also appear as amplitude modulation on the high resonant frequency of the pump. In general, there is low signal-to-noise ratio in low frequencies. Therefore we look into demodulation in high frequencies to find impulses. Several methods were applied on the signals object to 1) Separation of deterministic and non-deterministic parts of signal. 2) Separation of source signal and transmission path function. 3) Determine the optimal resonant frequency band and envelope analysis. In addition a new method to estimate the shaft speed from the vibration signal is proposed. Two segments were chosen from the old and new vibration signals for analysis. We used Minimum Entropy Deconvolution (MED) and Cepstrum Analysis to remove the transmission path effect from the vibration signal. We also used Time Synchronous Averaging (TSA), Autoregressive Filter (AR), Discrete/Randome Separation (DRS) and Cepstrum Analysis to separate deterministic components from non-deterministic components in the vibration signal. The latter resolves the first problem and former resolves the second one. Band-Pass Filtering, Wavelet Packet Transform (WPT), Hilbert-Huang Transform (HHT) and Spectral Kurtosis (SK) were used to detect excited resonant frequency for Demodulation and Envelope Analysis. High kurtosis is a measure of high impulsivity in a signal. Therefore all of our methods rely on kurtosis. The results demonstrate that the Minimum Entropy Deconvolution is a very effective method to remove the transmission path effect such that the defect impulses of the source can be clearly observed. A combination of wavelet and HHT as an Improved-HHT is a highly reliable method to detect the excited resonant frequencies. Spectral Kurtosis is an efficient and direct detector of the excited resonant frequencies. Cepstrum Editing or Liftering is a multi-purpose method for each stage. A new kind of cepstrum editing with a satisfactory output has been performed. Order Tracking is performed before Time Synchronous Averaging and Discrete/Random Separation and in this study there is no tachometer signals to order tracking. Therefore these methods were not effective. After examination on the old signal only the 5th and 27th harmonics of the shaft speed had a high amplitude. These frequencies are indeed the natural frequencies of the pump and do not indicate a fault. On the other hand, the new signal led to many peaks. The strongest of these peaks were the shaft speed harmonics and 1/2 harmonics. Lack of information on the pump specifications disables us to relate these peaks to specific faults. But these harmonics are generally presented in mechanical looseness.
Sun, 29 Nov 2015 00:00:00 GMThttp://hdl.handle.net/1956/116662015-11-29T00:00:00ZQuantifying uncertainties when monitoring marine environments in connection with geological storage of CO2http://hdl.handle.net/1956/11211
Quantifying uncertainties when monitoring marine environments in connection with geological storage of CO2
Hvidevold, Hilde Kristine
Doctoral thesis
Fri, 26 Jun 2015 00:00:00 GMThttp://hdl.handle.net/1956/112112015-06-26T00:00:00ZA robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous mediahttp://hdl.handle.net/1956/11048
A robust linearization scheme for finite volume based discretizations for simulation of two-phase flow in porous media
Radu, Adrian Florin; Nordbotten, Jan Martin; Pop, Iuliu Sorin; Kumar, Kundan
Journal article
In this work we consider a mathematical model for two-phase flow in porous media. The fluids are assumed immiscible and incompressible and the solid matrix non-deformable. The mathematical model for the two-phase flow is written in terms of the global pressure and a complementary pressure (obtained by using the Kirchhoff transformation) as primary unknowns. For the spatial discretization, finite volumes have been used (more precisely the multi-point flux approximation method) and in time the backward Euler method has been employed. We present here a new linearization scheme for the nonlinear system arising after the temporal and spatial discretization. We show that the scheme is linearly convergent. Numerical experiments are presented that sustain the theoretical results.
Tue, 01 Dec 2015 00:00:00 GMThttp://hdl.handle.net/1956/110482015-12-01T00:00:00ZNon-standard shocks in the Buckley-Leverett equationhttp://hdl.handle.net/1956/11045
Non-standard shocks in the Buckley-Leverett equation
Kalisch, Henrik; Mitrovic, Darko; Nordbotten, Jan Martin
Journal article
It is shown how delta shock waves which consist of Dirac delta distributions and classical shocks can be used to construct non-monotone solutions of the Buckley–Leverett equation. These solutions are interpreted using a recent variational definition of delta shock waves in which the Rankine–Hugoniot deficit is explicitly accounted for [6]. The delta shock waves are also limits of approximate solutions constructed using a recent extension of the weak asymptotic method to complex-valued approximations [15]. Finally, it is shown how these non-standard shocks can be fitted together to construct similarity and traveling-wave solutions which are non-monotone, but still admissible in the sense that characteristics either enter or are parallel to the shock trajectories.
Sat, 01 Aug 2015 00:00:00 GMThttp://hdl.handle.net/1956/110452015-08-01T00:00:00ZA Comparison of Anthropometric Measures for Assessing the Association between Body Size and Risk of Chronic Low Back Pain: TheHUNT Studyhttp://hdl.handle.net/1956/11019
A Comparison of Anthropometric Measures for Assessing the Association between Body Size and Risk of Chronic Low Back Pain: TheHUNT Study
Heuch, Ingrid; Heuch, Ivar; Hagen, Knut; Zwart, John-Anker
Journal article
<p>Background: Previous work indicates that overweight and obese individuals carry an increased risk of experiencing chronic low back pain (LBP). It is not known, however, how the association with body size depends on the choice of anthropometric measure used.</p>
<p>Objective: This work compares relationships with LBP for several measures of body size. Different results may indicate underlying mechanisms for the association between body size and risk of LBP.</p>
<p>Methods: In a cohort study, baseline information was collected in the community-based HUNT2 (1995&ndash;1997) and HUNT3 (2006&ndash;2008) surveys in Norway. Participants were 10,059 women and 8725 men aged 30&ndash;69 years without LBP, and 3883 women and 2662 men with LBP at baseline. Associations with LBP at end of follow-up were assessed by generalized linear modeling, with adjustment for potential confounders.</p>
<p>Results: Relationships between waist-hip-ratio and occurrence of LBP at end of follow-up were weak and non-significant after adjustment for age, education, work status, physical activity, smoking, lipid levels and blood pressure. Positive associations with LBP at end of follow-up were all significant for body weight, BMI, waist circumference and hip circumference after similar adjustment, both in women without and with LBP at baseline, and in men without LBP at baseline. After additional mutual adjustment for anthropometric measures, the magnitude of the association with body weight increased in women without LBP at baseline (RR: 1.130 per standard deviation, 95% CI: 0.995&ndash;1.284) and in men (RR: 1.124, 95% CI 0.976&ndash;1.294), with other measures showing weak associations only.</p>
<p>Conclusion: Central adiposity is unlikely to play a major role in the etiology of LBP. Total fat mass may be one common factor underlying the associations observed. The association with body weight remaining after mutual adjustment may reflect mechanical or structural components behind the relationship between overweight or obesity and LBP.</p>
Tue, 27 Oct 2015 00:00:00 GMThttp://hdl.handle.net/1956/110192015-10-27T00:00:00ZLayout of CCS monitoring infrastructure with highest probability of detecting a footprint of a CO2 leak in a varying marine environmenthttp://hdl.handle.net/1956/10990
Layout of CCS monitoring infrastructure with highest probability of detecting a footprint of a CO2 leak in a varying marine environment
Hvidevold, Hilde Kristine; Alendal, Guttorm; Johannessen, Truls; Ali, Alfatih Omer Mohammed Ahmed; Mannseth, Trond; Avlesen, Helge
Journal article
Monitoring of the marine environment for indications of a leak, or precursors of a leak, will be an intrinsic part of any subsea CO2 storage projects. A real challenge will be quantification of the probability of a given monitoring program to detect a leak and to design the program accordingly. The task complicates by the number of pathways to the surface, difficulties to estimate probabilities of leaks and fluxes, and predicting the fluctuating footprint of a leak. The objective is to present a procedure for optimizing the layout of a fixed array of chemical sensors on the seafloor, using the probability of detecting a leak as metric. A synthetic map from the North Sea is used as a basis for probable leakage points, while the spatial footprint is based on results from a General Circulation Model. Compared to an equally spaced array the probability of detecting a leak can be nearly doubled by an optimal placement of the available sensors. It is not necessarily best to place the first in the location of the highest probable leakage point, one sensor can monitor several potential leakage points. The need for a thorough baseline in order to reduce the detection threshold is shown.
Sat, 04 Apr 2015 00:00:00 GMThttp://hdl.handle.net/1956/109902015-04-04T00:00:00ZHeat Transfer Upscaling in Geothermal Reservoirshttp://hdl.handle.net/1956/10977
Heat Transfer Upscaling in Geothermal Reservoirs
Varzina, Anna
Master thesis
In this thesis we implement a numerical model of
heat transfer in geothermal reservoirs. We
use existing pressure and flow transport solvers
as a starting point to investigate discretization
techniques for a convection-conduction temperature
equation. Then we develop and analyse two
different heat transfer solvers: explicit and
implicit, that have different accuracy and
convergence requirements. For the convective part
of the energy equation the upwind scheme is
implemented and the two-point flux approximation
is used to discretize the conductive term. Usually
heat transfer simulations require large
computational time due to high resolution on a
fine scale. For efficient computation we
investigate flow-based upgridding techniques,
which were used before for fluid transport in
porous media. However upgridding and upscaling
can lead to less accurate results due to much loss
of details in a discrete model. We compare
solutions on different types of grids such as
Cartesian grid and flow-based grids that are
generated according to various indicators like
permeability, velocity, time-of-flight and thermal
conductivity. In this work we simulate an initial-
boundary value problem with a heat flow through
boundaries and try to investigate, which coarse
grid leads to the most accurate results when
solving the energy equation.
Fri, 20 Nov 2015 00:00:00 GMThttp://hdl.handle.net/1956/109772015-11-20T00:00:00ZDesign of a monitoring program in a varying marine environmenthttp://hdl.handle.net/1956/10952
Design of a monitoring program in a varying marine environment
Frøysa, Håvard Guldbrandsen
Master thesis
In this thesis we develop methods for optimal design of a monitoring program
for offshore geological CO2 storage. The goal is to find the layout of fixed
chemical sensors at the seafloor that maximizes the probability of detecting a
leakage. Numerical simulations of leakage scenarios are used as origin to predict
the regions that sensors monitor. Based on leakage scenarios, this gives the
detection probability. All methods are tested using test cases. The methods
could be applied to other problems involving monitoring of potential pollutants
into the ocean. The main results are inclusion of spatial variability in the
estimated leakage footprint and an exact inversion of the resulting footprint.
Mon, 14 Sep 2015 00:00:00 GMThttp://hdl.handle.net/1956/109522015-09-14T00:00:00ZA robust implicit scheme for two-phase flow in porous mediahttp://hdl.handle.net/1956/10951
A robust implicit scheme for two-phase flow in porous media
Kvashchuk, Anna
Master thesis
In this thesis we present a new implicit scheme
for the numerical simulation of two-phase flow
in porous media.
Linear finite elements are considered for the
spatial discretization.
The scheme is based on the iterative IMPES
approach and treats the capillary pressure term
implicitly to ensure stability. Under
assumption of smoothness of the capillary
pressure and the phase mobility curves, we were
able to prove convergence theorem for the
scheme.
Two dimensional numerical simulations
furthermore verify the convergence.
To illustrate the potential of the new scheme
we compare its computational efficiency to our
implementation of two other common approaches
to the problem:
IMPES and the fully implicit formulation solved
by Newton's method. The advantage of our scheme
over IMPES is improved stability for larger
time-step.
At the same time, it is cheaper in terms of
computational costs and memory requirements
compared to the Newton method.
Thu, 26 Nov 2015 00:00:00 GMThttp://hdl.handle.net/1956/109512015-11-26T00:00:00ZAn Approach for Investigation of Geochemical Rock-Fluid Interactionshttp://hdl.handle.net/1956/10903
An Approach for Investigation of Geochemical Rock-Fluid Interactions
Bringedal, Carina; Berre, Inga; Radu, Florin A.
Conference object
<p>Geochemistry has a substantial impact in exploiting of geothermal systems. When water is injected in a geothermal reservoir, the injected water and in-situ brine have different temperatures and chemical compositions and will flow through highly heterogeneous regions where the rock has varying chemical properties and where temperature and flow regimes can alter significantly.</p>
<p>As a consequence of flow and geochemical reactions, composition of reservoir fluids as well as reservoir rock properties will develop dynamically with time. Minerals dissolving and precipitating onto the reservoir matrix, can change the porosity and hence the permeability of the system substantially. Mineral solubility can change by the cooling of the rock by the injected water as well as by the injected water having a different salt content than the in-situ brine. The interaction between altering temperature, solute transport with mineral dissolution and precipitation, and fluid flow is highly coupled and challenging to model appropriately as the relevant physical processes jointly affect each other. The effect of changing porosity through the production period of the geothermal reservoir, may have severe impacts on operating conditions, as pores may close and block flow paths, or new pores may open to create enhanced flow conditions.</p>
<p>We propose an approach that models all three processes on a relevant time scale. The considered mathematical and corresponding overall numerical solution strategy enables us to investigate the coupling between flow, geochemical and thermal effects, as well as to develop tailored numerical approaches.</p>
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/1956/109032014-01-01T00:00:00ZInfluence of natural convection in a porous medium when producing from borehole heat exchangershttp://hdl.handle.net/1956/10902
Influence of natural convection in a porous medium when producing from borehole heat exchangers
Bringedal, Carina; Berre, Inga; Nordbotten, Jan Martin
Journal article
Convection currents in a porous medium form when the medium is subject to sufficient heating from below (or equivalently, cooling from above) or when cooled or heated from the side. In the context of geothermal energy extraction, we are interested in how the convection currents transport heat when a sealed borehole containing cold fluid extracts heat from the porous medium; also known as a borehole heat exchanger. Using pseudospectral methods together with domain decomposition, we consider two scenarios for heat extraction from a borehole; one system where the porous medium is initialized with constant temperature in the vertical direction and one system initialized with a vertical temperature gradient. We find the convection currents to have a positive effect on the heat extraction for the case with a constant initial temperature in the porous medium, and a negative effect for some of the systems with an initial temperature gradient in the porous medium: Convection gives a negative effect when the borehole temperature is close the initial temperature in the porous medium, but gradually provides a positive effect if the borehole temperature is decreased and the Rayleigh number is larger.
Thu, 01 Aug 2013 00:00:00 GMThttp://hdl.handle.net/1956/109022013-08-01T00:00:00ZLinear and nonlinear convection in porous media between coaxial cylindershttp://hdl.handle.net/1956/10901
Linear and nonlinear convection in porous media between coaxial cylinders
Bringedal, Carina; Berre, Inga; Nordbotten, Jan Martin; Rees, D. Andrew S.
Journal article
We uncover novel features of three-dimensional natural convection in porous media by investigating convection in an annular porous cavity contained between two vertical coaxial cylinders. The investigations are made using a linear stability analysis, together with high-order numerical simulations using pseudospectral methods to model the nonlinear regime. The onset of convection cells and their preferred planform are studied, and the stability of the modes with respect to different types of perturbation is investigated. We also examine how variations in the Rayleigh number affect the convection modes and their stability regimes. Compared with previously published data, we show how the problem inherits an increased complexity regarding which modes will be obtained. Some stable secondary modes or mixed modes have been identified and some overlapping stability regions for different convective modes are determined.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/1956/109012011-01-01T00:00:00ZModeling of heat transfer in porous media in the context of geothermal energy extractionhttp://hdl.handle.net/1956/10900
Modeling of heat transfer in porous media in the context of geothermal energy extraction
Bringedal, Carina
Doctoral thesis
<p>This dissertation concerns mathematical modeling related to heat transfer in the subsurface relevant for geothermal energy extraction. Two applications are considered: First, natural convection in a homogeneous porous medium around a borehole is studied. Secondly, pore scale models for non-isothermal reactive transport with changing porosity are formulated and upscaled.</p>
<p>Natural convection is the heat transfer due to currents arising from density differences in the fluid. A vertical borehole heat exchanger not injecting fluid into the subsurface, may encounter convection currents in the surrounding porous medium when the porous medium is saturated with water. These convection currents can affect the heat transfer into the borehole, hence have an influence on the heat production from the borehole. To address this issue, both a theoretical framework and a numerical approach for the natural convection is considered. In the theoretical framework a linear stability analysis is applied to quantify when and how convection currents may occur spontaneously without the presence of a producing borehole heat exchanger. A high-order numerical scheme is implemented to quantify the effect on an operating borehole. The linear stability analysis is performed for an idealized setting with a homogeneous porous medium filling an annular cylinder that is heated from below and cooled from above. The analysis provides criterions for onset of convection and the associated pattern for convection currents in the linearized case. As the onset criterion and the convection pattern that appears depend on the size of the annular cylinder, maps describing the onset criterion and the convection pattern as a function of the inner and outer radius of the annulus are created.</p>
<p>To investigate the non-linear regime of natural convection, pseudospectral methods are applied to discretize the non-linear model equations. Pseudospectral methods are highorder numerical schemes known for their exponential convergence rate. The convergence rate for our model problem is investigated, and the scheme is used to examine the stability of the convection currents found in the linear stability analysis. The linear and non-linear regimes are found to overlap when the convection is weak, while stronger convection introduces non-linear artifacts only visible through the simulations. Further, by applying a more realistic configuration for the geothermal reservoir, pseudospectral methods are used in combination with domain decomposition to estimate the effect from natural convection on an operating borehole. By varying the flow properties and temperature conditions, some cases where the presence of convection would lower the amount of heat that is produced by the borehole are found, while other cases give a positive effect on the heat production.</p>
<p>When geothermal energy is produced through injecting fluid into the subsurface and producing warmer fluid, other challenges arise. The in-situ groundwater and the injected water will typically have different temperature and ion content, hence the injection shifts the geochemical system from its original state of equilibrium. We focus on mineral precipitation and dissolution reactions induced by this shift. As most minerals have temperature-dependent solubilities, chemical reactions may be triggered by both the temperature change as well as the variations in ion content. Due to the rockfluid interactions mineral precipitation and dissolution entail, the porosity and hence the permeability of the porous medium can be changed, which again alters the production conditions for the geothermal reservoir. As the fluid flow, heat transport and reactive transport affect each other and involves processes over several time scales, this problem is highly coupled and challenging to model. This coupling is illustrated and used to motivate the need for pore scale models in order to better understand how the interactions between the physical processes behave at the pore scale. Further, upscaling through homogenization is performed on three different pore scale models, each applying different assumptions on the pore scale geometry or on the underlying physics. The upscaling process isolates the average behavior of the pore scale effects and shows how the processes are coupled at Darcy scale. One of the upscaled models is illustrated by implementing it using a finite volume scheme and is compared with some simpler models to illustrate when the upscaled model should be used and when it can be disregarded for a simpler model not honoring all the pore scale effects. Finite volume methods are conservative schemes, which are applied when consistent formulation of the transportation mechanisms is important.</p>
Fri, 02 Oct 2015 00:00:00 GMThttp://hdl.handle.net/1956/109002015-10-02T00:00:00ZThe Whitham Equation as a model for surface water waveshttp://hdl.handle.net/1956/10890
The Whitham Equation as a model for surface water waves
Moldabayev, Daulet; Kalisch, Henrik; Dutykh, Denys
Journal article
The Whitham equation was proposed as an alternate model equation for the simplified description of uni-directional wave motion at the surface of an inviscid fluid. As the Whitham equation incorporates the full linear dispersion relation of the water wave problem, it is thought to provide a more faithful description of shorter waves of small amplitude than traditional long wave models such as the KdV equation.
In this work, we identify a scaling regime in which the Whitham equation can be derived from the Hamiltonian theory of surface water waves. A Hamiltonian system of Whitham type allowing for two-way wave propagation is also derived. The Whitham equation is integrated numerically, and it is shown that the equation gives a close approximation of inviscid free surface dynamics as described by the Euler equations. The performance of the Whitham equation as a model for free surface dynamics is also compared to different free surface models: the KdV equation, the BBM equation, and the Padé (2,2) model. It is found that in a wide parameter range of amplitudes and wavelengths, the Whitham equation performs on par with or better than the three considered models.
Sat, 01 Aug 2015 00:00:00 GMThttp://hdl.handle.net/1956/108902015-08-01T00:00:00ZMatroider og lineære koderhttp://hdl.handle.net/1956/10780
Matroider og lineære koder
Larsen, Ann-Hege
Master thesis
Wed, 01 Jun 2005 00:00:00 GMThttp://hdl.handle.net/1956/107802005-06-01T00:00:00ZAlgebra i første klassehttp://hdl.handle.net/1956/10681
Algebra i første klasse
Lid, Elin Røkeberg
Master thesis
Oppgaven handler om algebra i første klasse. Hva er tidlig
algebra og hvordan kan vi undervise på en slik måte at vi
utvikler elevenes algebraiske tenkning? Tidlig algebra er ikke at
vi underviser skolealgebra tidligere, men det er en ny måte å
forholde seg til matematikkundervisning i småskolen på. Jeg har
gått gjennom eksisterende forskning og funnet konkrete
anbefalinger og undervisningsforløp. Dessuten har jeg tilpasset
andres opplegg og utviklet egne. Oppgaven inneholder dermed
anbefalinger for og eksempler på hvordan man kan undervise
algebraisk tenkning i første klasse.
Mon, 12 Oct 2015 00:00:00 GMThttp://hdl.handle.net/1956/106812015-10-12T00:00:00ZEnsemble Methods of Data Assimilation in Porous Media Flow for Non-Gaussian Prior Probability Densityhttp://hdl.handle.net/1956/10607
Ensemble Methods of Data Assimilation in Porous Media Flow for Non-Gaussian Prior Probability Density
Zhang, Yanhui
Doctoral thesis
<p>Ensemble-based data-assimilation methods have gained fast growth since the ensemble Kalman filter (EnKF) was introduced into the petroleum engineering. Many techniques have been developed to overcome the inherent disadvantages of the ensemble-based methods including the assumptions of linearity and Gaussianity, to make it more robust and reliable for practical reservoir applications. The current trend in petroleum reservoir history matching is towards taking into account more realistic reservoir models with complex geology. Geologic facies modeling plays an important role in the reservoir characterization as a way to reproduce important patterns of heterogeneity in petroleum reservoirs and to facilitate the modeling of petrophysical properties of reservoir rocks. Because the static data and general geological knowledge are almost always not sufficient to determine the distribution of geologic facies uniquely, it is advantageous to assimilate dynamic data to reduce the uncertainty.</p>
<p>The history-matching problem of geologic facies involves large nonlinearity and non-Gaussianity. When the ensemble Kalman filter (EnKF) or related ensemble-based methods are used to assimilate data in a straightforward way, the updated model variables would lose the geological realism and be contaminated with noise. Therefore, it is necessary to develop some effective measures to adapt the ensemble-based dataassimilation methods to the facies problems. This thesis is motivated by this necessity and explores the ways to conditioning geologic facies to production data using ensemble-based methods. The focus is placed on the post-processing approach.</p>
<p>By modifying the standard randomized maximum likelihood algorithm to accommodate non-Gaussian problems, we introduce a methodology that consists of a straightforward implementation of ensemble-based data assimilation in the first place and a sequential optimization procedure without iteration as a follow-up. In a similar manner, we develop another method for the post-processing of the updated reservoir models after data assimilation using ensemble-based methods. In the post-processing step, the objective function is composed of a weighted quadratic term measuring the distance to the posterior realizations, and a penalty term forcing model variables to take discrete values. A special emphasis is put on the investigation of the importance of the correlation information among the updated model variables introduced by the data which is usually ignored for the probability map approach. All of the proposed methodologies are evaluated via numerical experiments and demonstrated their utilities for improving the assimilation of data to geological facies models.</p>
<p>In this thesis, we also investigate the application of discrete curvelet transform in the denoising problem of updated model variables by the direct use of ensemble-based data-assimilation methods. According to the numerical experiments, the results show that curvelets are useful for denoising in the problem concerned but lose data match unless the covariance is included.</p>
Fri, 28 Aug 2015 00:00:00 GMThttp://hdl.handle.net/1956/106072015-08-28T00:00:00ZModels for CO2 injection with coupled thermal processeshttp://hdl.handle.net/1956/10591
Models for CO2 injection with coupled thermal processes
Kometa, Bawfeh Kingsley; Gasda, Sarah Eileen; Aavatsmark, Ivar
Journal article
Large-scale models of carbon dioxide (CO2) storage in geological formations must capture the relevant physical, chemical and thermodynamic processes that affect the migration and ultimate fate of injected CO2. These processes should be modeled over the appropriate length and time scales. Some important mechanisms include convection- driven dissolution, caprock roughness, and local capillary effects, all of which can impact the direction and speed of the plume as well as long-term trapping efficiency. In addition, CO2 can be injected at a different temperature than reservoir conditions, leading to significant density variation within the plume over space and time. This impacts buoyancy and migration patterns, which becomes particularly important for injection sites with temperature and pressure conditions near the critical point. Therefore, coupling thermal processes with fluid flow should be considered in order to correctly capture plume migration and trapping within the reservoir. This study focuses on compositional non- isothermal flow using 3D and vertically upscaled models. The model concept is demonstrated on simple systems. In addition, we explore CO2 thermodynamic models for reliable prediction of density under different injection pressures, temperatures and composition.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/1956/105912014-01-01T00:00:00ZAnalytical Methods for Upscaling of Fractured Geological Reservoirshttp://hdl.handle.net/1956/10558
Analytical Methods for Upscaling of Fractured Geological Reservoirs
Sævik, Pål Næverlid
Doctoral thesis
<p>Numerical simulations have become an essential tool for planning well operations in the subsurface, whether the application is petroleum production, geothermal energy, groundwater utilization or waste disposal and leakage. To represent a complicated heterogeneous subsurface reservoir in a simulator model, material properties within each part of the reservoir are replaced by effective properties through upscaling. In this thesis, analytical upscaling methods for fractured reservoirs are studied, with special emphasis on effective medium methods. These methods have been suggested for fracture upscaling by a number of authors, but reliable analytical error estimates and comparisons with comprehensive numerical simulations have been lacking.</p>
<p>The work in this thesis evaluates the accuracy of effective medium methods by providing an extensive comparison between analytical and numerical estimates, for both isotropic and anisotropic three-dimensional fracture configurations. The estimates are also analyzed with respect to known theoretical results, such as rigorous upper and lower bounds, asymptotic behavior and percolation properties.</p>
<p>It is found that one of the effective medium variants, the asymmetric self-consistent method, has the correct asymptotic behavior, satisfy all analytical bounds, and agree well with numerical percolation results. This is somewhat surprising, since the method has been regarded in the literature as having the weakest theoretical foundation. One explanation for the good results may be the special geometry that a fracture/matrix system represents, which agree well with the way the method is defined and derived.</p>
<p>As a part of the work in this thesis, effective medium formulations that are numerically stable for arbitrarily thin inclusions are developed. The formulations show good convergence properties in the general anisotropic case, and explicit expressions for the isotropic and slightly anisotropic case are also given. In the case of very thin inclusions, the new formulations allows the number of input parameters to be reduced.</p>
<p>Finally, the thesis investigates the use of assisted history matching for fractured reservoirs. It is shown that history matching of upscaled models may generate parameter distributions that are inconsistent with the underlying fracture description, result- ing in unphysical connectivity estimates for the fracture network. The problem can be avoided by history matching the fracture parameters directly, and include fracture upscaling as an integral part of the parameter inversion framework. This adds to the computational cost, but the additional computational effort is negligible if analytical fracture upscaling is used.</p>
Fri, 16 Oct 2015 00:00:00 GMThttp://hdl.handle.net/1956/105582015-10-16T00:00:00ZVertically averaged equations with variable density for CO2 flow in porous mediahttp://hdl.handle.net/1956/10548
Vertically averaged equations with variable density for CO2 flow in porous media
Andersen, Odd; Gasda, Sarah Eileen; Nilsen, Halvor Møll
Journal article
Carbon capture and storage has been proposed as a viable option to reduce CO 2 emissions. Geological storage of CO 2 where the gas is injected into geological formations for practically indefinite storage, is an integral part of this strategy. Mathematical models and numerical simulations are important tools to better understand the processes taking place underground during and after injection. Due to the very large spatial and temporal scales involved, commercial 3D-based simulators for the petroleum industry quickly become impractical for answering questions related to the long-term fate of injected CO 2 . There is an interest in developing simplified modeling tools that are effective for this type of problem. One approach investigated in recent years is the use of upscaled models based on the assumption of vertical equilibrium (VE). Under this assumption, the simulation problem is essentially reduced from 3D to 2D, allowing much larger models to be considered at the same computational cost. So far, most work on VE models for CO 2 storage has either assumed incompressible CO 2 or only permitted lateral variations in CO 2 density (semi-compressible). In the present work, we propose a way to fully include variable CO 2 density within the VE framework, making it possible to also model vertical density changes. We derive the fine-scale and upscaled equations involved and investigate the resulting effects. In addition, we compare incompressible, semi-compressible, and fully compressible CO 2 flow for some model scenarios, using an in-house, fully-implicit numerical code based on automatic differentiation, implemented using the MATLAB reservoir simulation toolkit.
Sun, 01 Mar 2015 00:00:00 GMThttp://hdl.handle.net/1956/105482015-03-01T00:00:00ZInternational Conference on Greenhouse Gas Technologies (GHGT-12) Models for CO2 injection with coupled thermal procehttp://hdl.handle.net/1956/10534
International Conference on Greenhouse Gas Technologies (GHGT-12) Models for CO2 injection with coupled thermal proce
Kometa, Bawfeh Kingsley; Gasda, Sarah Eileen; Aavatsmark, Ivar
Journal article
Large-scale models of carbon dioxide (CO2) storage in geological formations must capture the relevant physical, chemical and thermodynamic processes that affect the migration and ultimate fate of injected CO2. These processes should be modeled over the appropriate length and time scales. Some important mechanisms include convection- driven dissolution, caprock roughness, and local capillary effects, all of which can impact the direction and speed of the plume as well as long-term trapping efficiency. In addition, CO2 can be injected at a different temperature than reservoir conditions, leading to significant density variation within the plume over space and time. This impacts buoyancy and migration patterns, which becomes particularly important for injection sites with temperature and pressure conditions near the critical point. Therefore, coupling thermal processes with fluid flow should be considered in order to correctly capture plume migration and trapping within the reservoir. This study focuses on compositional non- isothermal flow using 3D and vertically upscaled models. The model concept is demonstrated on simple systems. In addition, we explore CO2 thermodynamic models for reliable prediction of density under different injection pressures, temperatures and composition.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/1956/105342014-01-01T00:00:00ZGood ethics or political and cultural censoring in science?http://hdl.handle.net/1956/10486
Good ethics or political and cultural censoring in science?
Torrissen, Ole; Glover, Kevin; Haug, tore; Misund, Ole Arve; Skaug, Hans J.; Kaiser, Matthias
Journal article
Peer-reviewed journals are the cornerstones to communicating scientific results. They play a crucial role in quality assurance through the review process, but they also create opportunities for discussions in the scientific community on the implications of the results or validation of methods and data. This requires that journals adhere to commonly accepted scientific standards and are open about their editorial policy. Norwegian scientists experience problems in getting research on minke whales accepted for publication where the data have been collected in association with commercial whaling. The journal Biology Letters refuses to publish papers based on data from the Norwegian whale register while publically claiming a sole focus on scientific quality. Although there are good arguments for claiming that clearly unethical research should not be rewarded with scientific publications, one also has to realize that some fields of research are beset with unresolved ethical and cultural debates. In these cases, it is to the benefit of the progress of science, and indeed society, to be open about the issues and support arguments through scientific studies. Political or cultural censoring of scientific information will in any case jeopardize the role of journals in quality assurance of scientific research and undermine the credibility of science as a supplier of objective and reliable knowledge.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/1956/104862012-01-01T00:00:00ZNumerical computation of a finite amplitude sound beamhttp://hdl.handle.net/1956/10468
Numerical computation of a finite amplitude sound beam
Berntsen, Jarle; Vefring, Erlend Heggelund
Working paper
Wed, 01 Jan 1986 00:00:00 GMThttp://hdl.handle.net/1956/104681986-01-01T00:00:00ZSolution strategies for nonlinear conservation lawshttp://hdl.handle.net/1956/10433
Solution strategies for nonlinear conservation laws
Skogestad, Jan Ole
Doctoral thesis
<p>Nonlinear conservation laws form the basis for models for a wide range of physical phenomena. Finding an optimal strategy for solving these problems can be challenging, and a good strategy for one problem may fail spectacularly for others. As different problems have different challenging features, exploiting knowledge about the problem structure is a key factor in achieving an efficient solution strategy.</p>
<p>Most strategies found in literature for solving nonlinear problems involve a linearization step, usually using Newton&#39;s method, which replaces the original nonlinear problem by an iteration process consisting of a series of linear problems. A large effort is then spent on finding a good strategy for solving these linear problems. This involves choosing suitable preconditioners and linear solvers. This approach is in many cases a good choice and a multitude of different methods have been developed.</p>
<p>However, the linearization step to some degree involves a loss of information about the original problem. This is not necessarily critical, but in many cases the structure of the nonlinear problem can be exploited to a larger extent than what is possible when working solely on the linearized problem. This may involve knowledge about dominating physical processes and specifically on whether a process is near equilibrium.</p>
<p>By using nonlinear preconditioning techniques developed in recent years, certain attractive features such as automatic localization of computations to parts of the problem domain with the highest degree of nonlinearities arise. In the present work, these methods are further refined to obtain a framework for nonlinear preconditioning that also takes into account equilibrium information. This framework is developed mainly in the context of porous media, but in a general manner, allowing for application to a wide range of problems. A scalability study shows that the method is scalable for challenging two-phase flow problems. It is also demonstrated for nonlinear elasticity problems.</p>
<p>Some models arising from nonlinear conservation laws are best solved using completely different strategies than the approach outlined above. One such example can be found in the field of surface gravity waves. For special types of nonlinear waves, such as solitary waves and undular bores, the well-known Korteweg-de Vries (KdV) equation has been shown to be a suitable model. This equation has many interesting properties not typical of nonlinear equations which may be exploited in the solver, and strategies usually reserved to linear problems may be applied. In this work includes a comparative study of two discretization methods with highly different properties for this equation.</p>
Fri, 21 Aug 2015 00:00:00 GMThttp://hdl.handle.net/1956/104332015-08-21T00:00:00ZDomain decomposition preconditioning for non-linear elasticity problemshttp://hdl.handle.net/1956/10432
Domain decomposition preconditioning for non-linear elasticity problems
Keilegavlen, Eirik; Skogestad, Jan Ole; Nordbotten, Jan Martin
Conference object
We consider domain decomposition techniques for a non-linear elasticity problem. Our main focus is on non-linear preconditioning, realized in the framework of additive Schwarz preconditioned inexact Newton (ASPIN) methods. The standard 1-level ASPIN method is extended to a 2-level method by adding a non-linear coarse solver. Numerical experiments show that the coarse component is necessary for scalability in terms of linear iterations inside the Newton loop. Moreover, for problems that are dominated by nonlinearities that are not localized in space the non-linear coarse iterations are crucial for achieving computational efficiency.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/1956/104322014-01-01T00:00:00ZModellering og estimering av romlig avhengighet i forsikringhttp://hdl.handle.net/1956/10416
Modellering og estimering av romlig avhengighet i forsikring
Sellereite, Nikolai
Master thesis
De seneste årene har det blitt publisert flere studier hvor bayesianske hierarkiske modeller, med gitte spatiale avhengighetsstrukturer, blir foreslått som potensielle verktøy i forsikringsselskapers arbeid rettet mot geografisk prisdifferensiering. I denne oppgaven blir problemstillingen angrepet fra et frekventisk ståsted, hvor fokuset er begrenset til modeller for antall skader. Skadestørrelse, levetid o.l. er andre eksempler på responsvariabler hvor det teoretiske rammeverket kan anvendes. Parameterestimeringen blir utført ved hjelp av maksimum likelihood, og som følge av høydimensjonale integral introduseres Laplace-approksimasjon og automatisk derivasjon, hvor estimeringsprosedyren automatiseres ved hjelp av pakken Template Model Builder (TMB). Den latente spatiale effekten modelleres som Gaussian Markov random fields (GMRFs) med ulike valg av spatial avhengighetsstruktur. Modeller med og uten latente spatiale variabler tilpasses en simulert forsikringsportefølje, hvor modellene uten latente spatiale variabler tilsvarer generaliserte lineære modeller. Valideringen av den prediktive evnen til modellene blir utført ved å simulere 1000 nye forsikringsporteføljer.
Mon, 01 Jun 2015 00:00:00 GMThttp://hdl.handle.net/1956/104162015-06-01T00:00:00ZMønstre og kvalitet. Analyse av samtaler i matematikkundervisninghttp://hdl.handle.net/1956/10360
Mønstre og kvalitet. Analyse av samtaler i matematikkundervisning
Mjaavatn, Geir
Master thesis
Oppgaven diskuterer hvordan man kan analysere samtaler mellom lærer og elever i matematikkundervisning. Deretter diskuterer oppgaven hva som kjennetegner en kvalitativ god samtale.
Wed, 03 Jun 2015 00:00:00 GMThttp://hdl.handle.net/1956/103602015-06-03T00:00:00ZStatistiske modelleringer i det europeiske gassmarkedethttp://hdl.handle.net/1956/10346
Statistiske modelleringer i det europeiske gassmarkedet
Frontéri, Anette Ullestad
Master thesis
I denne oppgaven studerer vi naturgassmarkedet
i Europa. Først ser vi på hvilke forhold som
påvirker gassprisen. Deretter studerer vi
statistisk teori om tidsrekker, volatilitet og
copula. Til slutt blir det gjort statistiske
modelleringer på naturgasspriser i tre
europeiske gassmarkeder. Naturgassprisene blir
omformet til log-avkastninger. GARCH-modeller
blir brukt for å modellere volatiliteten i log-
avkastningene. Copula blir brukt til å
modellere avhengighetsstrukturen, som er mellom
de marginale fordelingene i de europeiske
gassmarkedene. Til slutt blir det brukt
kointegrasjon for å beskrive sammenhengen
mellom de europeiske gassprisene.
Mon, 01 Jun 2015 00:00:00 GMThttp://hdl.handle.net/1956/103462015-06-01T00:00:00ZNatural Convection in Layered Porous Media between Coaxial Cylindershttp://hdl.handle.net/1956/10336
Natural Convection in Layered Porous Media between Coaxial Cylinders
Selheim, Bianca
Master thesis
In this thesis a mathematical model is developed to describe the onset of natural convection in a two-layer porous medium located between two coaxial cylinders, motivated by natural convection processes in geothermal systems.
The cylinders are heated from below and cooled from above. We consider the top and bottom to be impermeable and perfectly heat conducting, while
the sidewalls are assumed to be impermeable and insulated. At the interface between the layers, we require continuity in temperature, pressure, vertical flow and heat flow.
We apply linear stability analysis to determine the criterion for onset of natural convection in the bottom layer. An investigation of the effects of permeability contrasts between the two layers on the critical Rayleigh number is performed. We present new results, as our analysis applies to a two-layer medium in cylindrical coordinates. The results are validated by comparison with similar previous studies for a single-layer medium with the same geometry, and for a layered medium within a box geometry. The model has real
life applications as it may work as an indicator of the presence of natural convection in the subsurface and since the results can be applied in benchmarking of a numerical simulator.
Sun, 31 May 2015 00:00:00 GMThttp://hdl.handle.net/1956/103362015-05-31T00:00:00ZDetecting atmospheric rivers using persistent homologyhttp://hdl.handle.net/1956/10335
Detecting atmospheric rivers using persistent homology
Alfsvåg, Kristian Stusdal
Master thesis
This master's thesis is a first investigation on the problem of seeing whether it is possible to detect atmospheric rivers using persistent homology. Two different computations are done and a basic analysis is made. In addition an implementation of persistent homology made during the thesis is described.
Mon, 01 Jun 2015 00:00:00 GMThttp://hdl.handle.net/1956/103352015-06-01T00:00:00ZLogarithmic Hochschild homologyhttp://hdl.handle.net/1956/10333
Logarithmic Hochschild homology
Piceghello, Stefano
Master thesis
The purpose of this thesis is to analyse the logarithmic Hochschild homology for pre-log rings and to provide some tools to compute it in certain cases. One of the main strategies that we will employ to describe the log Hochschild homology will entail passing through the module of the log Kahler differentials. An additional technique that we can adopt to gather information about the log Hochschild homology of some specific pre-log rings is to interlock it in a long exact sequence, relating it to the ordinary Hochschild homology groups. An example in which this method applies nicely is the case where the remaining terms of the long exact sequence are the Hochschild homology of polynomial algebras in a finite number of variables, for which we try to provide an exhaustive description.
Mon, 01 Jun 2015 00:00:00 GMThttp://hdl.handle.net/1956/103332015-06-01T00:00:00ZNumerical Modelling of Microbial Enhanced Oil Recovery with Focus on Dynamic Effects: An Iterative Approachhttp://hdl.handle.net/1956/10262
Numerical Modelling of Microbial Enhanced Oil Recovery with Focus on Dynamic Effects: An Iterative Approach
Skiftestad, Kai
Master thesis
Recovering more of the available oil has been a main driver behind the extensive work done in the field of enhanced oil recovery (EOR) over the last decades. Microbial en- hanced oil recovery (MEOR) has been heavily researched, and is picking up pace com- pared with other EOR methods used today. MEOR is economically attractive and has a huge potential if applied in accordance to reservoir conditions. This thesis considers a two-phase flow regime in homogeneous porous media, under the influence of microbial activity. The mathematical model includes the concept of dynamic capillary pressure, and is based on Darcy's law, the principle of mass conservation and the diffusion/dispersion-advection equation. The inclusion of the dynamic capillary pressure makes this model classified as a so-called non-standard model. In this work we aim to explore this, as well as the effect microbes have on flow, and ultimately oil production. Implementation of the mathematical model has been done in MATLAB by using a new, fully implicit, iterative approach, to cope with the fact that the dynamic capillarity induces an additional temporal derivative in the two-phase model. The spatial dis- cretization has been carried out with the use of a control volume method, the TPFA, on a cell-centered grid in one dimension. The scheme is related to the papers [1-3]. The effects of dynamic capillary pressure are shown to be small at the macroscale for realistic oil reservoirs, while clearly visible in an extreme case which have been set up. Regarding microbial activity we have constructed relations between concentration and interfacial tension based on the work in [4-8]. This is done to model the effect of reduced fluid-fluid tension on flow and further the oil production. It is shown that a substantial concentration of microbes have a positive effect on the production, while small concentrations do not differ significantly from the case of no concentration.
Sat, 27 Jun 2015 00:00:00 GMThttp://hdl.handle.net/1956/102622015-06-27T00:00:00ZApplication of Vertically-Integrated Models with Subscale Vertical Dynamics to Field Sites for CO2 Sequestrationhttp://hdl.handle.net/1956/10095
Application of Vertically-Integrated Models with Subscale Vertical Dynamics to Field Sites for CO2 Sequestration
Guo, B; Bandilla, Karl W.; Keilegavlen, Eirik; Doster, Florian; Celia, Michael A.
Journal article
<p>To address the engineering questions on security issues of geological carbon sequestration (GCS), a broad range of computational models with different levels of complexity have been developed - from multiphase multicomponent fully coupled three-dimensional models to simplified analytical solutions. Within this wide range of models, a family of so-called vertical equilibrium (VE) models has been developed. These models assume that CO2 and brine have segregated due to buoyancy and reached a hydrostatic pressure distribution in the vertical direction. Such VE models are computationally efficient due to the dimensional reduction in the vertical, and accurate as long as the VE assumption is satisfied. However, a study comparing results from a VE model to results from a full three- dimensional model found that there are realistic conditions for which the VE assumption is not justified, especially for geological formations that have low vertical permeability, on the order of 10 milliDarcy or lower [1].
</p><p>
In an attempt to overcome the VE limitation of the vertically-integrated models, a new type of vertically-integrated model which relaxes the VE assumption while still using a vertically-integrated framework has been developed [2]. The new vertically-integrated model is cast into a multiscale framework, where the coarse scale is the horizontal domain and the fine scale is the vertical domain corresponding to the thickness of a geologic formation. This type of model maintains much of the computational advantages of the VE models, while allowing a much wider range of problem to be modelled. In this paper, we extend the model in [2] to include horizontally layered geologic heterogeneities and develop a new dynamic reconstruction model, which we refer to as a “multi-layer dynamic reconstruction” model. The model in [2] is called “single-layer dynamic reconstruction” model to distinguish the modelling approaches. We apply both dynamic reconstruction models to field injection sites, including a hypothetical injection scenario into the Mount Simon formation in the Illinois Basin, USA and the well-known industrial-scale injection into the Utsira formation at the Sleipner site in Norway. The modelling results show that the multi-layer dynamic reconstruction model is capable of dealing with horizontally layered heterogeneities and gives results that agree reasonably well with results from the full multi-dimensional model, although in geologic layers with high permeability the reconstruction algorithm is not able to fully capture the horizontal driving forces due to buoyancy. This could be important over long time scales for highly permeable geologic layers. The single- layer dynamic reconstruction model was shown be to the right model choice for homogeneous formations with relatively low permeability where it takes a long time for CO2 and brine to reach vertical equilibrium [2]. In this study, we found that for homogeneous formations with high permeability and steep capillary pressure curves, the single-layer dynamic reconstruction model gives results that are analogous to vertical equilibrium models, with the fast segregation dynamics requiring small time steps in the dynamic reconstruction algorithm. As such vertical equilibrium models are the appropriate choice for systems with high permeability, although we also note that the behaviour of the brine relative permeability curve at low brine saturations is an additional important consideration and must be included in the overall analysis.</p>
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/1956/100952014-01-01T00:00:00ZAssociation between body height and chronic low back pain: a follow-up in the Nord-Trøndelag Health Studyhttp://hdl.handle.net/1956/10052
Association between body height and chronic low back pain: a follow-up in the Nord-Trøndelag Health Study
Heuch, Ingrid; Heuch, Ivar; Hagen, Knut; Zwart, John-Anker
Journal article
<p>Objective: To study potential associations between body height and subsequent occurrence of chronic low back pain (LBP).</p>
<p>Design: Prospective cohort study.</p>
<p>Setting: The North-Tr&oslash;ndelag Health Study (HUNT). Data were obtained from a whole Norwegian county in the HUNT2 (1995&ndash;1997) and HUNT3 (2006&ndash;2008) surveys.</p>
<p>Participants: Altogether, 3883 women and 2662 men with LBP, and 10 059 women and 8725 men without LBP, aged 30&ndash;69 years, were included at baseline and reported after 11 years whether they suffered from LBP.</p>
<p>Main outcome measure: Chronic LBP, defined as pain persisting for 3 months during the previous year.</p>
<p>Results: Associations between body height and risk and recurrence of LBP were evaluated by generalised linear modelling. Potential confounders, such as BMI, age, education, employment, physical activity, smoking, blood pressure and lipid levels were adjusted for. In women with no LBP at baseline and body height &ge;170 cm, a higher risk of LBP was demonstrated after adjustment for other risk factors (relative risk 1.19, 95% CI 1.03 to 1.37; compared with height &lt;160 cm). No relationship was established among men or among women with LBP at baseline.</p>
<p>Conclusions: In women without LBP, a body height &ge;170 cm may predispose to chronic LBP 11 years later. This may reflect mechanical issues or indicate a hormonal influence.</p>
Mon, 15 Jun 2015 00:00:00 GMThttp://hdl.handle.net/1956/100522015-06-15T00:00:00ZSkipped spawning in Northeast Arctic haddock Melanogrammus aeglefinushttp://hdl.handle.net/1956/10051
Skipped spawning in Northeast Arctic haddock Melanogrammus aeglefinus
Skjæraasen, Jon Egil; Korsbrekke, Knut; Nilsen, Trygve; Fonn, Merete; Kjesbu, Olav Sigurd; Dingsør, Gjert Endre; Nash, Richard David Marriott
Journal article
Large interannual fluctuations in the numbers of offspring joining a teleost population are common, yet factors affecting offspring production, a key driver of fish population size and demography, are often poorly understood. For some iteroparous teleosts, spawning omission (‘skipping’) following sexual maturation may occur, but this is typically difficult to verify. Through the detection of post-ovulatory follicles, i.e. evidence of past spawning activity, in gonads of females not spawning in the current year, we demonstrate skipping in Northeast Arctic (NEA) haddock Melanogrammus aeglefinus. Based on samples obtained just prior to the main spawning season in the Barents Sea from February to April in 2009 to 2012, the estimated population frequency of skippers ranged from 23 (2009) to 64% (2011) for females ≥35 cm in total length found in this area at this time. Skipping was associated with limited energy reserves and persisted with age, although it appeared to be most common in 5 yr old females. This suggests that skipping is linked to a combination of feeding condition and demography of the fish population. While previously virtually undocumented in this species, this phenomenon appears to be an integral life history feature for NEA haddock and may have a major impact on the annual realised egg production in this population. Finally, given the similarity between the results reported here and those recently published for NEA cod, we postulate that skipping may be a common occurrence in gadoids undertaking long, energetically demanding spawning migrations.
Wed, 22 Apr 2015 00:00:00 GMThttp://hdl.handle.net/1956/100512015-04-22T00:00:00ZBrøkforståing hjå elevar som startar i den vidaregåande skulenhttp://hdl.handle.net/1956/9949
Brøkforståing hjå elevar som startar i den vidaregåande skulen
Antun, Jorunn
Master thesis
Målet for denne oppgåva har vore å få større
innsikt i kva brøkforståing elevar som startar
i den vidaregåande skulen kan ha. To
forskingsspørsmål har lagt til grunn for
arbeidet mitt:
1.Kva forståing av og dugleikar i brøk og
brøkrekning kan ein finna hjå elevar som
startar i den vidaregåande skulen?
2.Kva misoppfatningar rundt omgrepet brøk og
brøkrekning kan ein finna hjå elevane?
Trettito elevar har delteke i undersøkinga. Dei
har nett starta i den vidaregåande skulen,
etter å ha gått på ulike ungdomsskular spreidd
over heile landet. Ein skriftleg
kartleggingstest gav informasjon om kva elevane
meistra innan sentrale aspekt ved brøk og
moglege misoppfatningar knytt til brøk og
brøkrekning. Deretter vart om lag ein tredel av
informantane intervjua for å få større innsikt
i korleis dei tenkjer. Datamaterialet vart
analysert ut frå eit konstruktivistisk syn på
læring.
Undersøkinga mi viser eit stort spenn med omsyn
til brøkforståing. Utfordringar kjem i særleg
grad til synes i overgangane mellom kjennskap
til prosedyrar, iverksetjing av prosedyrar og
forståing for korleis desse faktisk verkar.
Nokre elevar har berre fragmentarisk innsikt i
prosedyrar. Andre elevar har eit isolert fokus
på reglar; dei er kjend med ulike prosedyrar og
nyttar algoritmar ved operasjonar på brøk. Feil
som kjem fram er knytt til faktisk utføring av
prosedyrane; ulike reglar vert blanda saman,
eller det stoppar opp for elevar når ei
prosedyre ikkje kan nyttast direkte. Ein ser
vidare døme på elevar som evnar å sjå
samanhengar, og som i praksis greier å setja
saman ulike prosedyrar og nytta brøkomgrepet i
ulike kontekstar. Mange greier likevel fortsatt
ikkje forklara kva dei gjer og kvifor dei gjer
det dei gjer. Mangel på forståing ser ut til å
vera den gjennomgåande faktoren som set grenser
for meistring på høgare nivå. Samla sett får
ein her stadfesta at eit grunnlag bygd på
dugleikar utan forståing er laust fundamentert;
det skal lite til før elevane vert usikre eller
at reknereglar vert gløymd.
Manglande forståing gjer at ulike
misoppfatningar kjem til syne, både når brøkar
skal ordnast og samanliknast, når ekvivalente
brøkar skal lagast og ved operasjonar på brøk.
Undersøkinga viser m.a. at fleire elevar
overfører kunnskap om bruk av dei naturlege
tala til brøk og brøkrekning, og at elevar
strevar med å sjå og bruka ideen om ekvivalente
brøkar i ein større samanheng. Dei avdekka
misoppfatningane gjev samla eit vidare innsyn i
grunnleggjande manglar knytt til brøkforståing
hjå elevar som startar i den vidaregåande
skulen.
Thu, 30 Apr 2015 00:00:00 GMThttp://hdl.handle.net/1956/99492015-04-30T00:00:00ZEntropy solutions of the compressible Euler equationshttp://hdl.handle.net/1956/9868
Entropy solutions of the compressible Euler equations
Svärd, Magnus
Journal article
We consider the three-dimensional Euler equations of gas dynamics
on a bounded periodic domain and a bounded time interval. We prove that
Lax-Friedrichs scheme can be used to produce a sequence of solutions with
ever finer resolution for any appropriately bounded (but not necessarily small)
initial data. Furthermore, we show that if the density remains strictly positive
in the sequence of solutions at hand, a subsequence converges to an entropy
solution. We provide numerical evidence for these results by computing a
sensitive Kelvin-Helmholtz problem.
Submitted to BIT Numerical Mathematics.
Mon, 11 May 2015 00:00:00 GMThttp://hdl.handle.net/1956/98682015-05-11T00:00:00ZSUCCESS: SUbsurface CO2 storage–Critical elements and superior strategyhttp://hdl.handle.net/1956/9794
SUCCESS: SUbsurface CO2 storage–Critical elements and superior strategy
Aker, Eyvind; Bjørnarå, Tore Ingvald; Braathen, Alvar; Brandvoll, Øyvind; Dahle, Helge K.; Nordbotten, Jan Martin; Aagaard, Per; Hellevang, Helge; Alemu, Binyam Lema; Pham, Van Thi Hai; Johansen, Harald; Wangen, Magnus; Nøttvedt, Arvid; Aavatsmark, Ivar; Johannessen, Truls; Durand, Dominique
Journal article
SUCCESS is a Center for Environmental Energy Research in Norway and performs research related to geological storage of CO2 in the subsurface. The SUCCESS centre is established by the Research Council of Norway together with several Norwegian research institutes and universities. The centre is hosted by Christian Michelsen Research. Through international cooperation and open research the SUCCESS centre will fill gaps in strategic knowledge and provide a system for learning and development of new competency to ensure safe and effective CO2 injection, storage and monitoring. In this paper we briefly present the main focus areas of the centre and some recent results obtained by the research partners. The results relate to geochemical effects, reservoir modeling, monitoring the geomechanical respond and the marine environment. A brief status on the field trial, Longyearbyen CO2 Lab, at Svalbard is also provided.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/1956/97942011-01-01T00:00:00ZField-case simulation of CO2-plume migration using vertical-equilibrium modelshttp://hdl.handle.net/1956/9791
Field-case simulation of CO2-plume migration using vertical-equilibrium models
Nilsen, Halvor Møll; Herrera, Paulo; Ashraf, Seyed Meisam; Ligaard, Ingeborg S.; Iding, Martin; Hermanrud, Christian; Lie, Knut-Andreas; Nordbotten, Jan Martin; Dahle, Helge K.; Keilegavlen, Eirik
Journal article
When injected in deep saline aquifers, CO2 moves radially away from the injection well and progressively higher in the formation because of buoyancy forces. Analyzes have shown that after the injection period, CO2 will potentially migrate over several kilometers in the horizontal direction but only tens of meters in the vertical direction, limited by the aquifer caprock. Because of the large horizontal plume dimensions, three-dimensional numerical simulations of the plume migration over long periods of time are computationally intensive. Thus, to get results within a reasonable time frame, one is typically forced to use coarse meshes and long time steps which result in inaccurate results because of numerical errors in resolving the plume tip.
Given the large aspect ratio between the vertical and horizontal plume dimensions, it is reasonable to approximate the CO2 migration using vertically averaged models. Such models can, in many cases, be more accurate than coarse three-dimensional computations. In particular, models based on vertical equilibrium (VE) are attractive to simulate the long-term fate of CO2 sequestered into deep saline aquifers. The reduced spatial dimensionality resulting from the vertical integration ensures that the computational performance of VE models exceeds the performance of standard three-dimensional models. Thus, VE models are suitable to study the long-time and large-scale behavior of plumes in real large-scale CO2-injection projects. We investigate the use of VE models to simulate CO2 migration in a real large-scale field case based on data from the Sleipner site in the North Sea. We discuss the potential and limitations of VE models and show how VE models can be used to give reliable estimates of long-term CO2 migration. In particular, we focus on a VE formulation that incorporates the aquifer geometry and heterogeneity, and that considers the effects of hydrodynamic and residual trapping. We compare the results of VE simulations with standard reservoir simulation tools on test cases and discuss their advantages and limitations and show how, provided that certain conditions are met, they can be used to give reliable estimates of long-term CO2 migration.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/1956/97912011-01-01T00:00:00ZUpscaled modeling of CO2 injection and migration with coupled thermal processeshttp://hdl.handle.net/1956/9747
Upscaled modeling of CO2 injection and migration with coupled thermal processes
Gasda, Sarah Eileen; Stephansen, Annette; Aavatsmark, Ivar; Dahle, Helge K.
Journal article
A practical modeling approach for CO2 storage over relatively large length and time scales is the vertical-equilibrium
model, which solves partially integrated conservation equations for flow in two lateral dimensions. We couple heat
transfer within the vertical equilibrium framework for fluid flow, focusing on thermal processes that most impact the
CO2 plume. We investigate a simplified representation of heat exchange that also includes transport of heat within
the plume. In addition, we explore available CO2 thermodynamic models for reliable prediction of density under
different injection pressures and temperatures. The model concept is demonstrated on simplified systems.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/1956/97472013-01-01T00:00:00ZAssessing Model Uncertainties Through Proper Experimental Designhttp://hdl.handle.net/1956/9745
Assessing Model Uncertainties Through Proper Experimental Design
Hvidevold, Hilde Kristine; Alendal, Guttorm; Johannessen, Truls; Mannseth, Trond
Journal article
This paper assesses how parameter uncertainties in the model for rise velocity of CO2 droplets in the ocean cause uncertainties in their rise and dissolution in marine waters. The parameter uncertainties in the rise velocity for both hydrate coated and hydrate free droplets are estimated from experiment data. Thereafter the rise velocity is coupled with a mass transfer model to simulate the fate of dissolution of a single droplet.
The assessment shows that parameter uncertainties are highest for large droplets. However, it is also shown that in some circumstances varying the temperature gives significant change in rise distance of droplets.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/1956/97452013-01-01T00:00:00ZRisk of Leakage versus Depth of Injection in Geological Storagehttp://hdl.handle.net/1956/9724
Risk of Leakage versus Depth of Injection in Geological Storage
Celia, Michael A.; Nordbotten, Jan Martin; Bachu, Stefan; Dobossy, Mark E.; Court, Benjamin
Journal article
One of the outstanding challenges for large-scale CCS operations is to develop reliable quantitative risk assessments with a focus on leakage of both injected CO2 and displaced brine. A critical leakage pathway is associated with the century-long legacy of oil and gas exploration and production, which has led to many millions of wells being drilled. Many of those wells are in locations that would otherwise be excellent candidates for CCS operations, especially across many parts of North America. Quantitative analysis of the problem requires special computational techniques because of the unique challenges associated with simulation of injection and leakage in systems that include hundreds or thousands of existing wells over domains characterized by layered structures in the vertical direction and very large horizontal extent. An important feature of these kinds of systems is the depth of each well, and the fact that the number of wells penetrating different formations decreases as a function of depth. As such, one might reasonably expect the risk of leakage to decrease with depth of injection. With the special computational models developed to simulate injection and leakage along multiple wells, in layered systems with multiple formations, quantitative assessment of risk reduction as a function of injection depth can be made. An example of such a system corresponds to the Wabamun Lake area southwest of Edmonton, Alberta, Canada, where several large coal-fired power plants are located. Use of information about both the existing wells and the local stratigraphy allows a realistic model to be constructed. Leakage along existing wells is assumed to follow Darcy’s Law, and is characterized by a set of effective permeability values. These values are assigned stochastically, using several different methods, within a Monte Carlo simulation framework. Computational results show the clear trade-off between depth of injection and risk of leakage. The results also show how properties within the different formations affect the risk profiles. In the Wabamun Lake area, one of the formations has the highest injectivity, by far, while having a moderate number of existing wells. Its moderate risk of leakage, as compared to injections in formations above and below, shows some of the key factors that are likely to influence injection design for large-scale CCS operations.
Sun, 01 Feb 2009 00:00:00 GMThttp://hdl.handle.net/1956/97242009-02-01T00:00:00ZActive and integrated management of water resources throughout CO2 capture and sequestration operationshttp://hdl.handle.net/1956/9720
Active and integrated management of water resources throughout CO2 capture and sequestration operations
Court, Benjamin; Celia, Michael A.; Nordbotten, Jan Martin; Elliot, Thomas R.
Journal article
Most projected climate change mitigation strategies will require a significant expansion of CO2 Capture and Sequestration (CCS) in the next two decades. Four major categories of challenges are being actively researched: CO2 capture cost, geological sequestration safety, legal and regulatory barriers, and public acceptance. Herein we propose an additional major challenge category across all CCS operations: Water management. For example a coal-fired power plant retrofitted for CCS requires twice as much cooling water as the original plant. This increased demand may be accommodated by brine extraction and treatment, which would concurrently function as large-scale pressure management and a potential source of freshwater. At present the interactions among freshwater extraction, CO2 injection, and brine management are being considered too narrowly -in the case of freshwater almost completely overlooked- in the technical and regulatory CCS community. This paper presents an overview of each of these challenges and potential integration opportunities. Active management of CCS operations through an integrated approach -including brine production, treatment, use for cooling, and partial reinjection- can address challenges simultaneously with several synergistic advantages. The paper also considers the related potential impacts of pore space competition (with future groundwater use, gas storage and shale gas) on CCS expansion. Freshwater and brine must become key decision making inputs throughout CCS operations, building on existing successful industrial-scale integrations.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/1956/97202011-01-01T00:00:00ZAn efficient software framework for performing industrial risk assessment of leakage for geological storage of CO2http://hdl.handle.net/1956/9719
An efficient software framework for performing industrial risk assessment of leakage for geological storage of CO2
Dobossy, Mark E.; Celia, Michael A.; Nordbotten, Jan Martin
Journal article
In response to anthropogenic CO2 emissions, geological storage has emerged as a practical and scalable bridge technology while renewables and other environmentally friendly energy production methods mature. While an attractive solution, geological storage of CO2 has inherent risk. Two primary concerns are recognized: (1) leakage of CO2 through caprock imperfections, and (2) brine displacement resulting in contamination of drinking water sources. Three mechanisms for both CO2 and brine leakage have been identified: diffuse leakage through the caprock, leakage through faults and fractures in the caprock, and finally, leakage through man-made pathways such as abandoned wells from oil and gas exploration. While the first two leakage mechanisms are important, we emphasize the risks associated with the presence of abandoned wells. This is due to the large number and density of wells from a history of oil and gas exploration around the world, and the high degree of uncertainty surrounding the properties of these abandoned wells. With current proposed legislation in both the United States and Europe, a need is emerging for practical assessment of leakage risk. In order to accurately predict leakage of brine and CO2 from the injection layer, the geological information for the injection site and the location and makeup of the man-made leakage pathways previously alluded to must be taken into account. Unfortunately, both the geology and abandoned well metadata are typically high in uncertainty, which must be accounted for. With such a high number of random variables, the current state of the art is running many realizations of a system, using a Monte Carlo approach. This requires that the underlying solution algorithms be accurate, and efficient. In the past, many researchers in both academia and industry have turned to robust numerical analysis packages used in the oil industry. However, due to the large range of scales important to this problem (domains of tens of kilometers on a side affected by leakage pathways with diameters of tens of centimeters) such modeling techniques become computationally expensive for all but the most basic analysis. A computational model developed at Princeton University, and currently being commercialized by Geological Storage Consultants, LLC has been shown to be efficient with sufficient accuracy to allow for comprehensive risk assessment of CO2 injection projects. The model allows for mixing solution methods- using computationally expensive algorithms for formations of greater importance (e.g.- the injection formation) and more efficient, simplified algorithms in other areas of the domain. This ability to arbitrarily mix solution methods offers significant flexibility in the design and execution of models. This paper addresses the framework and algorithms used, and illustrates the importance of efficiency and parallelism using the case study of an injection site in Alberta, Canada. We show how the framework can be used for project planning, for risk mitigation (insurance), and for regulatory groups. Finally, the importance of flexible analysis tools that allow for efficient and effective management of computational resources is discussed.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/1956/97192011-01-01T00:00:00ZDetecting leakage of brine or CO2 through abandoned wells in a geological sequestration operation using pressure monitoring wellshttp://hdl.handle.net/1956/9718
Detecting leakage of brine or CO2 through abandoned wells in a geological sequestration operation using pressure monitoring wells
Nogues, Juan P.; Nordbotten, Jan Martin; Celia, Michael A.
Journal article
For risk assessment, policy design and GHG emission accounting it is extremely important to know if any CO2 or brine has leaked from a geological sequestration (GS) operation. As such, it is important to understand if it is possible to use certain technologies to detect it. This detection of leakage is one of the most challenging problems associated with GS due to the high uncertainty in the nature and location of leakage pathways. In North America for example millions of legacy oil and gas wells present the possibility of CO2 and brine to leak out of the injection formation. The available information for these potential leaky wells is very limited and the main parameters that control leakage, like permeability of the sealing material are not known. Here we propose to explore the possibility of detecting such leakage by the use of pressure-monitoring wells located in a formation overlying the injection formation. The detection analysis is based on a system of equations that solve for the propagation of a pressure pulse using the superposition principle and an approximation to the well function. We explore the questions of what can be gained by using pressure-monitoring wells and what are the limitations given a specific accuracy threshold of the measuring device. We also try to answer the question of where these monitoring wells should be placed to optimize the objective of a monitoring scheme. We believe these results can ultimately lead to practical design strategies for monitoring schemes, including quantitative estimation of increased probability of leak detection per added observation well.
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/1956/97182011-01-01T00:00:00ZThe impact of local-scale processes on large-scale CO2 migration and immobilizationhttp://hdl.handle.net/1956/9717
The impact of local-scale processes on large-scale CO2 migration and immobilization
Gasda, Sarah Eileen; Nordbotten, Jan Martin; Celia, Michael A.
Journal article
<p>Storage security of injected carbon dioxide (CO2) is an essential component of risk management for geological carbon sequestration operations. During the injection and early post-injection periods, CO2 leakage may occur along faults and leaky wells, but this risk may be partly managed by proper site selection and sensible deployment of monitoring and remediation technologies. On the other hand, long-term storage security is an entirely different risk management problem—one that is dominated by a mobile CO2 plume that may travel over very large spatial distances, over long time periods, before it is trapped by a variety of different physical and chemical processes. In the post-injection phase, the mobile CO2 plume migrates in large part due to buoyancy forces, following the natural topography of the geological formation. The primary trapping mechanisms are capillary and solubility trapping, which evolve over thousands to tens of thousands of years and can immobilize a significant portion of the mobile, free-phase CO2 plume. However, both the migration and trapping processes are inherently complex, involving a combination of small and large spatial scales and acting over a range of time scales. Solubility trapping is a prime example of this complexity, where small-scale density instabilities in the dissolved CO2 region leads to convective mixing that has that has a significant effect on the large-scale dissolution process over very long time scales. Another example is the effect of capillary forces on the evolution of mobile CO2, an often-neglected process except with regard to residual trapping. As the plume migrates due to buoyancy and viscous forces, local capillary effects acting at the CO2-brine interface lead to a transition zone where both fluids are present in the mobile state. This small-scale effect may have a significant impact on large-scale plume migration as well as long-term residual and dissolution trapping. Using appropriate models that can capture both large and small-scale effects is essential for understanding the role of these processes on the long-term storage security of CO2 sequestration operations.</p>
<p>There are several approaches to modeling long-term CO2 trapping mechanisms. One modeling option is the use of traditional numerical methods, which are often highly sophisticated models that can handle multiple complex phenomena with high levels of accuracy. However, these complex models quickly become prohibitively expensive for the type of large-scale, long-term modeling that is necessary for risk assessment applications such as the late post-injection period. We present an alternative modeling option that combines vertically-averaged governing equations with an upscaled representation of the dissolution-convective mixing process and the local capillary transition zone at the CO2-brine interface. CO2 injection is solved numerically on a coarse grid, capturing the large-scale injection problem and the post-injection capillary trapping, while the upscaled dissolution and capillary fringe models capture these subscale effects and eliminate the need for expensive grid refinement to capture the subscale instabilities associated with convective mixing or the details of the capillary transition zone. With this modeling approach, we demonstrate the effect of different modeling choices associated with dissolution and capillary processes for typical large-scale geological systems.</p>
Sat, 01 Jan 2011 00:00:00 GMThttp://hdl.handle.net/1956/97172011-01-01T00:00:00ZA 3D computational study of effective medium methods applied to fractured mediahttp://hdl.handle.net/1956/9705
A 3D computational study of effective medium methods applied to fractured media
Sævik, Pål Næverlid; Berre, Inga; Jakobsen, Morten; Lien, Martha
Journal article
This work evaluates and improves upon existing effective medium methods for permeability upscaling in fractured media. Specifically, we are concerned with the asymmetric self-consistent, symmetric self-consistent, and differential methods. In effective medium theory, inhomogeneity is modeled as ellipsoidal inclusions embedded in the rock matrix. Fractured media correspond to the limiting case of flat ellipsoids, for which we derive a novel set of simplified formulas. The new formulas have improved numerical stability properties, and require a smaller number of input parameters. To assess their accuracy, we compare the analytical permeability predictions with three-dimensional finite-element simulations. We also compare the results with a semi-analytical method based on percolation theory and curve-fitting, which represents an alternative upscaling approach. A large number of cases is considered, with varying fracture aperture, density, matrix/fracture permeability contrast, orientation, shape, and number of fracture sets. The differential method is seen to be the best choice for sealed fractures and thin open fractures. For high-permeable, connected fractures, the semi-analytical method provides the best fit to the numerical data, whereas the differential method breaks down. The two self-consistent methods can be used for both unconnected and connected fractures, although the asymmetric method is somewhat unreliable for sealed fractures. For open fractures, the symmetric method is generally the more accurate for moderate fracture densities, but only the asymmetric method is seen to have correct asymptotic behavior. The asymmetric method is also surprisingly accurate at predicting percolation thresholds.
Tue, 01 Oct 2013 00:00:00 GMThttp://hdl.handle.net/1956/97052013-10-01T00:00:00ZDetermining individual variation in growth and its implication for life-history and population processes using the Empirical Bayes methodhttp://hdl.handle.net/1956/9682
Determining individual variation in growth and its implication for life-history and population processes using the Empirical Bayes method
Vincenzi, Simone; Mangel, Marc; Crivelli, Alain J.; Munch, Stephan; Skaug, Hans J.
Journal article
The differences in demographic and life-history processes between organisms living in the same population have important
consequences for ecological and evolutionary dynamics. Modern statistical and computational methods allow the
investigation of individual and shared (among homogeneous groups) determinants of the observed variation in growth. We
use an Empirical Bayes approach to estimate individual and shared variation in somatic growth using a von Bertalanffy
growth model with random effects. To illustrate the power and generality of the method, we consider two populations of
marble trout Salmo marmoratus living in Slovenian streams, where individually tagged fish have been sampled for more
than 15 years. We use year-of-birth cohort, population density during the first year of life, and individual random effects as
potential predictors of the von Bertalanffy growth function’s parameters k (rate of growth) and L∞ (asymptotic size). Our
results showed that size ranks were largely maintained throughout marble trout lifetime in both populations. According to
the Akaike Information Criterion (AIC), the best models showed different growth patterns for year-of-birth cohorts as well as
the existence of substantial individual variation in growth trajectories after accounting for the cohort effect. For both
populations, models including density during the first year of life showed that growth tended to decrease with increasing
population density early in life. Model validation showed that predictions of individual growth trajectories using the
random-effects model were more accurate than predictions based on mean size-at-age of fish.
Thu, 11 Sep 2014 00:00:00 GMThttp://hdl.handle.net/1956/96822014-09-11T00:00:00ZInvestigating population genetic structure in a highly mobile marine organism: The minke whale Balaenoptera acutorostrata acutorostrata in the North East Atlantichttp://hdl.handle.net/1956/9675
Investigating population genetic structure in a highly mobile marine organism: The minke whale Balaenoptera acutorostrata acutorostrata in the North East Atlantic
Sanchez, Maria Quintela; Skaug, Hans J.; Øien, Nils Inge; Haug, Tore; Seliussen, Bjørghild Breistein; Solvang, Hiroko Kato; Pampoulie, Christophe; Kanda, Naohisa; Pastene, Luis A.; Glover, Kevin
Journal article
Inferring the number of genetically distinct populations and their levels of connectivity is of key importance for the
sustainable management and conservation of wildlife. This represents an extra challenge in the marine environment where
there are few physical barriers to gene-flow, and populations may overlap in time and space. Several studies have
investigated the population genetic structure within the North Atlantic minke whale with contrasting results. In order to
address this issue, we analyzed ten microsatellite loci and 331 bp of the mitochondrial D-loop on 2990 whales sampled in
the North East Atlantic in the period 2004 and 2007–2011. The primary findings were: (1) No spatial or temporal genetic
differentiations were observed for either class of genetic marker. (2) mtDNA identified three distinct mitochondrial lineages
without any underlying geographical pattern. (3) Nuclear markers showed evidence of a single panmictic population in the
NE Atlantic according STRUCTURE’s highest average likelihood found at K = 1. (4) When K = 2 was accepted, based on the
Evanno’s test, whales were divided into two more or less equally sized groups that showed significant genetic
differentiation between them but without any sign of underlying geographic pattern. However, mtDNA for these
individuals did not corroborate the differentiation. (5) In order to further evaluate the potential for cryptic structuring, a set
of 100 in silico generated panmictic populations was examined using the same procedures as above showing genetic
differentiation between two artificially divided groups, similar to the aforementioned observations. This demonstrates that
clustering methods may spuriously reveal cryptic genetic structure. Based upon these data, we find no evidence to support
the existence of spatial or cryptic population genetic structure of minke whales within the NE Atlantic. However, in order to
conclusively evaluate population structure within this highly mobile species, more markers will be required.
Tue, 30 Sep 2014 00:00:00 GMThttp://hdl.handle.net/1956/96752014-09-30T00:00:00ZNumerical solutions of two-phase flow with applications to CO2 sequestration and polymer floodinghttp://hdl.handle.net/1956/9666
Numerical solutions of two-phase flow with applications to CO2 sequestration and polymer flooding
Mykkeltvedt, Trine Solberg
Doctoral thesis
<p>This thesis addresses challenges related to mathematical and numerical modeling of flow in porous media. To address these challenges, two applications are considered: firstly, counter-current two-phase flow in a heterogeneous porous media and secondly, polymer flooding in the context of enhanced oil recovery. Furthermore, an upscaled model for CO2 migration is used to estimate effective rates of convective mixing from commercial-scale injection.</p>
<p>Numerically, the upstream mobility scheme is widely used to solve hyperbolic conservation laws. For flow in heterogeneous porous media there exists no convergence analysis for this scheme. Studies of the convergence performance of this scheme are important due to the extensive use of the upstream mobility scheme in the reservoir simulation community. We show that the upstream mobility scheme may exhibit large errors compared to the physically relevant solution when applied to a counter-current flow in a reservoir where discontinuities in the flux function are introduced through the permeability. A small perturbation of the relative permeability values can lead to a large difference in the solution produced by the upstream mobility scheme. Not only does the scheme encounter large errors compared to what is considered to be the physically relevant solution, but the solution also lacks entropy consistency.</p>
<p>High-resolution schemes are often used for model problems where high accuracy is required in the presence of shocks or discontinuities. Polymer flooding represents such a system and is a difficult process to model, especially since the dynamics of the flow lead to concentration fronts that are not self-sharpening. The application of modern high-resolution schemes to a system that models polymer flooding is considered and different first- and higher-order schemes are compared in terms of how the discontinuities are treated. Through numerous numerical experiments some special numerical artifacts of the polymer system are uncovered. The need of high-resolution schemes and the importance of their applicability for the polymer problem is addressed.</p>
<p>The process of CO2 migration ranges over multiple scales and results in challenges when it comes to modeling and simulation of this system. This expresses the need for a upscaled model and upscaled parameters that can capture both large and smallscale spatial and temporal effects. The ongoing CO2-injection at the Utsira formation is considered as a field-scale study for CO2 storage. Through an upscaled model for CO2 migration we get the first field-scale estimates of the effective upscaled convective mixing rates in this context. The findings are comparable but somewhat higher than reported in the existing literature based on fine-scale numerical simulations. Our work validates the use of numerical simulations to obtain upscaled convective mixing rates, while at the same time validating that convective mixing is an important quantifiable storage mechanism at the Utsira formation. To account for uncertainties in the description of the storage formation, sensitivity studies are conducted relative to some of the most uncertain parameters.</p>
Fri, 19 Dec 2014 00:00:00 GMThttp://hdl.handle.net/1956/96662014-12-19T00:00:00ZDo abnormal serum lipid levels increase the risk of chronic low back pain? The Nord-Trøndelag Health Studyhttp://hdl.handle.net/1956/9614
Do abnormal serum lipid levels increase the risk of chronic low back pain? The Nord-Trøndelag Health Study
Heuch, Ingrid; Heuch, Ivar; Hagen, Knut; Zwart, John-Anker
Journal article
Background: Cross-sectional studies suggest associations between abnormal lipid levels and prevalence of low back pain (LBP), but it is not known if there is any causal relationship.
Objective: The objective was to determine, in a population-based prospective cohort study, whether there is any relation between levels of total cholesterol, high density lipoprotein (HDL) cholesterol and triglycerides and the probability of experiencing subsequent chronic (LBP), both among individuals with and without LBP at baseline.
Methods: Information was collected in the community-based HUNT 2 (1995–1997) and HUNT 3 (2006–2008) surveys of an entire Norwegian county. Participants were 10,151 women and 8731 men aged 30–69 years, not affected by chronic LBP at baseline, and 3902 women and 2666 men with LBP at baseline. Eleven years later the participants indicated whether they currently suffered from chronic LBP.
Results: Among women without LBP at baseline, HDL cholesterol levels were inversely associated and triglyceride levels positively associated with the risk of chronic LBP at end of follow-up in analyses adjusted for age only. Adjustment for the baseline factors education, work status, physical activity, smoking, blood pressure and in particular BMI largely removed these associations (RR: 0.96, 95% CI: 0.85–1.07 per mmol/l of HDL cholesterol; RR: 1.16, 95% CI: 0.94–1.42 per unit of lg(triglycerides)). Total cholesterol levels showed no associations. In women with LBP at baseline and men without LBP at baseline weaker relationships were observed. In men with LBP at baseline, an inverse association with HDL cholesterol remained after complete adjustment (RR: 0.83, 95% CI: 0.72–0.95 per mmol/l).
Conclusion: Crude associations between lipid levels and risk of subsequent LBP in individuals without current LBP are mainly caused by confounding with body mass. However, an association with low HDL levels may still remain in men who are already affected and possibly experience a higher pain intensity.
Thu, 18 Sep 2014 00:00:00 GMThttp://hdl.handle.net/1956/96142014-09-18T00:00:00ZIdentification of subsurface structures using electromagnetic data and shape priorshttp://hdl.handle.net/1956/9612
Identification of subsurface structures using electromagnetic data and shape priors
Tveit, Svenn; Bakr, Shaaban A.; Lien, Martha; Mannseth, Trond
Journal article
We consider the inverse problem of identifying large-scale subsurface structures using the controlled source
electromagnetic method. To identify structures in the subsurface where the contrast in electric conductivity
can be small, regularization is needed to preserve structural information. We propose to combine
two approaches for regularization of the inverse problem. In the first approach we utilize a model-based,
reduced, composite representation of the electric conductivity that is highly flexible, even for a moderate
number of degrees of freedom. With a low number of parameters, the inverse problem is efficiently solved
using a standard, second-order gradient-based optimization algorithm. Further regularization is obtained
using structural prior information, available, e.g., from interpreted seismic data. The reduced conductivity
representation is suitable for incorporation of structural prior information. Such prior information can,
however, not be accurately modeled with a gaussian distribution. To alleviate this, we incorporate the
structural information using shape priors. The shape prior technique requires the choice of kernel function,
which is application dependent. We argue for using the conditionally positive definite kernel which is shown
to have computational advantages over the commonly applied gaussian kernel for our problem. Numerical
experiments on various test cases show that the methodology is able to identify fairly complex subsurface
electric conductivity distributions while preserving structural prior information during the inversion.
Sun, 01 Mar 2015 00:00:00 GMThttp://hdl.handle.net/1956/96122015-03-01T00:00:00ZInversion of CSEM Data for Subsurface Structure Identification and Numerical Assessment of the Upstream Mobility Schemehttp://hdl.handle.net/1956/9611
Inversion of CSEM Data for Subsurface Structure Identification and Numerical Assessment of the Upstream Mobility Scheme
Tveit, Svenn
Doctoral thesis
<p>Part I</p>
<p>In this part of the thesis, two different methodologies for solving the inverse problem of mapping the subsurface electric conductivity distribution using controlled source electromagnetic (CSEM) data are presented. The two inversion methodologies are based on a classical and a Bayesian approach for solving inverse problems, respectively.</p>
<p>In the classical approach, we regularize the inverse problem by incorporating structural prior information available from, e.g., interpreted seismic data. In many cases, the outcome of an interpretation of seismic data cannot be well approximated by a Gaussian distribution. Hence, to incorporate non-Gaussian prior information we have applied the shape prior technique. Here, an implicit transformation of variables facilitates the incorporation of non-Gaussian prior information, at the expense of an applicationdependent kernel function.</p>
<p>In the Bayesian approach, a combination of prior knowledge and observed data results in a solution given as a posterior probability density function (PDF). To sample from the posterior PDF, a sequential Bayesian method, the ensemble Kalman filter (EnKF), is applied. Structural prior information is naturally incorporated as a part of the Bayesian framework.</p>
<p>To represent large-scale subsurface structures two model-based, composite parameterizations based on the level-set representation are applied in the inversion methodologies. By using a reduced number of parameters in the representation, a regularization of the inverse problem is achieved. Moreover, it enables the use of second-order gradientbased optimization algorithms in the classical approach.</p>
<p>Part II</p>
<p>In this part of the thesis, a numerical investigation of the upstream mobility scheme for calculating fluid flow in porous media is presented. Previous studies have shown that the upstream mobility scheme experienced erroneous behaviour when approximating pure gravity segregation flow in 1D heterogeneous porous media. The errors shown, however, were small in magnitude. In this work, numerical experiments, where we include both advection and gravity segregation, are conducted. It is shown that the errors produced in this case may be larger in magnitude than for pure gravity segregation, but are only found for countercurrent flow situations.</p>
Fri, 27 Mar 2015 00:00:00 GMThttp://hdl.handle.net/1956/96112015-03-27T00:00:00ZInexact linear solvers for control volume discretizations in porous mediahttp://hdl.handle.net/1956/9588
Inexact linear solvers for control volume discretizations in porous media
Keilegavlen, Eirik; Nordbotten, Jan Martin
Journal article
We discuss the construction ofmulti-level inexact
linear solvers for control volume discretizations for porous
media. The methodology forms a contrast to standard iterative
solvers by utilizing an algebraic hierarchy of approximations
which preserve the conservative structure of the
underlying control volume. Our main result is the generalization
of multiscale control volume methods as multilevel
inexact linear solvers for conservative discretizations
through the design of a particular class of preconditioners.
This construction thereby bridges the gap between
multiscale approximation and linear solvers. The resulting
approximation sequence is referred to as inexact solvers.
We seek a conservative solution, in the sense of controlvolume
discretizations, within a prescribed accuracy. To this
end, we give an abstract guaranteed a posteriori error bound
relating the accuracy of the linear solver to the underlying
discretization. These error bounds are explicitly computable
for the grids considered herein. The afore-mentioned hierarchy
of conservative approximations can also be considered
in the context of multi-level upscaling, and this perspective
is highlighted in the text as appropriate. The new
construction is supported by numerical examples highlighting
the performance of the inexact linear solver realized
in both a multi- and two-level context for two- and threedimensional
heterogeneous problems defined on structured
and unstructured grids. The numerical examples assess the
performance of the approach both as an inexact solver, as
well in comparison to standard algebraic multigrid methods.
Wed, 03 Dec 2014 00:00:00 GMThttp://hdl.handle.net/1956/95882014-12-03T00:00:00ZCapturing the coupled hydro-mechanical processes occurring during CO2 injection – example from In Salahhttp://hdl.handle.net/1956/9489
Capturing the coupled hydro-mechanical processes occurring during CO2 injection – example from In Salah
Bjørnarå, Tore Ingvald; Mathias, Simon A.; Nordbotten, Jan Martin; Park, Joonsang; Bohloli, Bahman
Journal article
At In Salah, CO2 is removed from the production stream of several natural gas fields and re-injected into a deep and relatively thin
saline formation, in three different locations. The observed deformation on the surface above the injection sites have partly been
contributed to expansion and compaction of the storage aquifer, but analysis of field data and measurements from monitoring has
verified that substantial activation of fractures and faults occur. History-matching observed data in numerical models involve
several model iterations at a high computation cost. To address this, a simplified model that captures the key hydro-mechanical
effects, while retaining a reasonable accuracy when applied to realistic field data from In Salah, has been derived and compared to
a fully resolved model. Results from the case study presented here show a significant saving in computational cost (36%) and a
computational speed-up factor of 2.7.
Wed, 01 Jan 2014 00:00:00 GMThttp://hdl.handle.net/1956/94892014-01-01T00:00:00ZPseudo H-type algebras and sub-Riemannian cut locushttp://hdl.handle.net/1956/9449
Pseudo H-type algebras and sub-Riemannian cut locus
Autenried, Christian
Doctoral thesis
Fri, 13 Feb 2015 00:00:00 GMThttp://hdl.handle.net/1956/94492015-02-13T00:00:00ZOn weak solutions and a covergent numerical scheme for the compressible Navier-Strokes Equationshttp://hdl.handle.net/1956/9442
On weak solutions and a covergent numerical scheme for the compressible Navier-Strokes Equations
Svärd, Magnus
Journal article
In this paper, the three-dimensional compressible Navier-Stokes
equations are considered on a periodic domain. We propose a semi-discrete
numerical scheme and derive a priori bounds that ensures that the resulting
system of ordinary di erential equations is solvable for any h > 0. An a
posteriori examination that density remain uniformly bounded away from 0
will establish that a subsequence of the numerical solutions converges to a
weak solution of the compressible Navier-Stokes equations.
Thu, 26 Feb 2015 00:00:00 GMThttp://hdl.handle.net/1956/94422015-02-26T00:00:00ZAssessment of Sequential and Simultaneous Ensemble-based History Matching Methods for Weakly Non-linear Problemshttp://hdl.handle.net/1956/9398
Assessment of Sequential and Simultaneous Ensemble-based History Matching Methods for Weakly Non-linear Problems
Fossum, Kristian
Doctoral thesis
<p>The ensemble Kalman filter (EnKF) has, since its introduction in 1994, gained much attention as a tool for sequential data assimilation in many scientific areas. In recent years, the EnKF has been utilized for estimating the poorly known petrophysical parameters in petroleum reservoir models. The ensemble based methodology has inspired several related methods, utilized both in data assimilation and for parameter estimation. All these methods, including the EnKF, can be shown to converge to the correct solution in the case of a Gaussian prior model, Gaussian data error, and linear model dynamics. However, for many problems, where the methods are applied, this is not satisfied. Moreover, several numerical studies have shown that, for such cases, the different methods have different approximation error.</p>
<p>Considering parameter estimation for problems where the model depends on the parameters in a non-linear fashion, this thesis explore the similarities and differences between the EnKF and the alternative methods. Several characteristics are established, and it is shown that each method represents a specific combination of these characteristics. By numerical comparison, it is further shown that a variation of the characteristics produce a variation of the approximation error.</p>
<p>A special emphasis is put on the effect of one characteristic, whether data are assimilated sequentially or simultaneously. Typically, several data types are utilized in the parameter estimation problem. In this thesis, we assume that each data depends on the parameters in a specific non-linear fashion. Considering the assimilation of two weakly non-linear data with different degree of non-linearity, we show, through analytical studies, that the difference between sequential and simultaneous assimilation depends on the combination of data.</p>
<p>Via numerical modelling, we investigate the difference between sequential and simultaneous assimilation on toy models and simplified reservoir problems. Utilizing realistic reservoir data, we show that the assumption of difference in non-linearity for different data holds. Moreover, we demonstrate that, for favourable degree of nonlinearity, it is beneficial to assimilate the data ordered after ascending degree of nonlinearity.</p>
Fri, 13 Feb 2015 00:00:00 GMThttp://hdl.handle.net/1956/93982015-02-13T00:00:00ZDynamic Capillary Effects in the Simulation of Flow and Transport in Porous Media: A New Linearisation Methodhttp://hdl.handle.net/1956/9374
Dynamic Capillary Effects in the Simulation of Flow and Transport in Porous Media: A New Linearisation Method
Teveldal, Silje Kjønaas
Master thesis
In this thesis mathematical models with and without dynamic capillary effects are developed to model water flow and solute transport through a porous medium. The system of equations are discretised using the finite volume method TPFA in space and the backward Euler method in time. To solve the nonlinear systems appearing at each time step numerically, robust linearisation methods are proposed. These methods do not involve the computation of derivatives. The methods are analysed and have been shown to be linearly convergent and robust. Moreover, the convergence was shown to be independent of mesh size. The influence that the dynamic effects have on flow and transport is studied numerically. Additional numerical experiments were conducted to study the convergence of the linearisation schemes. The numerical results are shown to be in correspondence with the theoretical results.
Thu, 25 Dec 2014 00:00:00 GMThttp://hdl.handle.net/1956/93742014-12-25T00:00:00ZOn the extension giving the truncated Witt vectorshttp://hdl.handle.net/1956/9347
On the extension giving the truncated Witt vectors
Skjøtskift, Torgeir
Master thesis
We explore the theory of cohomology of groups and
the classification of group extensions with abelian
kernel. We then look at the group extensions that underlie
the truncated Witt vectors on the truncation set {1,p}
where p is a prime number. It turns out that we
can do without the multiplicative structure on the
source ring A by factoring the extension's representing
cocycle through a map into the p-fold tensor
product of A divided out by the C_p-action.
Mon, 05 Jan 2015 00:00:00 GMThttp://hdl.handle.net/1956/93472015-01-05T00:00:00ZTensor Induction as Left Kan Extensionhttp://hdl.handle.net/1956/9319
Tensor Induction as Left Kan Extension
Aye, Kaythi
Master thesis
Tue, 02 Dec 2014 00:00:00 GMThttp://hdl.handle.net/1956/93192014-12-02T00:00:00ZWeak solutions and convergent numerical schemes of Brenner-Navier-Stokes equationshttp://hdl.handle.net/1956/9230
Weak solutions and convergent numerical schemes of Brenner-Navier-Stokes equations
Svärd, Magnus
Research report
Lately, there has been some interest in modifications of the compressible
Navier-Stokes equations to include diffusion of mass. In this paper,
we investigate possible ways to add mass diffusion to the 1-D Navier-Stokes
equations without violating the basic entropy inequality. As a result, we recover
a general form of Brenner's modification of the Navier-Stokes equations.
We consider Brenner's system along with another modification where the viscous
terms collapse to a Laplacian diffusion. For each of the two modifications,
we derive a priori estimates for the PDE, suffciently strong to admit a weak
solution; we propose a numerical scheme and demonstrate that it satisfies the
same a priori estimates. For both modifications, we then demonstrate that the
numerical schemes generate solutions that converge to a weak solution (up to
a subsequence) as the grid is refined.
Tue, 20 Jan 2015 00:00:00 GMThttp://hdl.handle.net/1956/92302015-01-20T00:00:00ZModeling of oil reservoirs with focus on microbial induced effectshttp://hdl.handle.net/1956/9207
Modeling of oil reservoirs with focus on microbial induced effects
Babatunde, Stanley Owulabi
Master thesis
As an abstract to this thesis, we review some literatures in EOR and discussed the processes, strength and weakness of Microbial Enhanced Oil Recovery techniques. A two phase flow model
comprising water and oil via the concept of mean pressure has been formulated using mass conservation equations, Darcy's law and constitutive relations. This resulted in a set of coupled nonlinear parabolic partial
differential equation with primary variables being the mean pressure and water saturation. We discretized these equations in one dimension using a control volume discretization scheme in space and implicit Euler
in time. We employed the IMPES approach which decoupled the primary variables. A model validation test was made by comparison with an analytical solution and with the Couplex-Gas benchmark. The model was used to
investigate two major mechanisms by which the activities of bacterial helps in enhancing the recovery of the residual oil.
Sat, 01 Nov 2014 00:00:00 GMThttp://hdl.handle.net/1956/92072014-11-01T00:00:00ZMathematical modelling of slow drug release from collagen matriceshttp://hdl.handle.net/1956/9181
Mathematical modelling of slow drug release from collagen matrices
Erichsen, Birgitte Riisøen
Master thesis
This master's thesis is about controlled drug release, which is a relatively new area of mathematical modelling. In this thesis there have been two major focuses. The first is to further understand the model for drug release from collagen matrices developed earlier by solving it with a different numerical scheme, and the second to develop a new model based on a different geometry. Both models are based on mass conservation and Fick's law, and are therefore possible to compare. The two models have been discretized and implemented, and the results compared to experimental data.
Tue, 23 Sep 2014 00:00:00 GMThttp://hdl.handle.net/1956/91812014-09-23T00:00:00ZCell-centered finite volume discretizations for deformable porous mediahttp://hdl.handle.net/1956/9014
Cell-centered finite volume discretizations for deformable porous media
Nordbotten, Jan Martin
Journal article
The development of cell-centered finite volume discretizations for deformation is motivated by the desire for a compatible approach with the discretization of fluid flow in deformable porous media. We express the conservation of momentum in the finite volume sense, and introduce three approximations methods for the cell-face stresses. The discretization method is developed for general grids in one to three spatial dimensions, and leads to a global discrete system of equations for the displacement vector in each cell, after which the stresses are calculated based on a local expression. The method allows for anisotropic, heterogeneous and discontinuous coefficients.
The novel finite volume discretization is justified through numerical validation tests, designed to investigate classical challenges in discretization of mechanical equations. In particular our examples explore the stability with respect to the Poisson ratio and spatial discontinuities in the material parameters. For applications, logically Cartesian grids are prevailing, and we also explore the performance on perturbations on such grids, as well as on unstructured grids. For reference, comparison is made in all cases with the lowest-order Lagrangian finite elements, and the finite volume methods proposed herein is comparable for approximating displacement, and is superior for approximating stresses.
Mon, 11 Aug 2014 00:00:00 GMThttp://hdl.handle.net/1956/90142014-08-11T00:00:00ZExplicit volume-preserving splitting methods for divergence-free ODEs by tensor-product basis decompositionshttp://hdl.handle.net/1956/9006
Explicit volume-preserving splitting methods for divergence-free ODEs by tensor-product basis decompositions
Munthe-Kaas, Antonella Zanna
Journal article
We discuss the construction of volume-preserving splitting methods based on a tensor product of single-variable basis functions. The vector field is decomposed as the sum of elementary divergence-free vector fields (EDFVFs), each of them corresponding to a basis function. The theory is a generalization of the monomial basis approach introduced in Xue & Zanna (2013, BIT Numer. Math., 53, 265–281) and has the trigonometric splitting of Quispel & McLaren (2003, J. Comp. Phys., 186, 308–316) and the splitting in shears of McLachlan & Quispel (2004, BIT, 44, 515–538) as special cases. We introduce the concept of diagonalizable EDFVFs and identify the solvable ones as those corresponding to the monomial basis and the exponential basis. In addition to giving a unifying view of some types of volume-preserving splitting methods already known in the literature, the present approach allows us to give a closed-form solution also to other types of vector fields that could not be treated before, namely those corresponding to the mixed tensor product of monomial and exponential (including trigonometric) basis functions.
Sun, 23 Feb 2014 00:00:00 GMThttp://hdl.handle.net/1956/90062014-02-23T00:00:00ZSymmetric spaces and Lie triple systems in numerical analysis of differential equationshttp://hdl.handle.net/1956/9002
Symmetric spaces and Lie triple systems in numerical analysis of differential equations
Munthe-Kaas, Hans; Quispel, Reinout; Zanna, Antonella
Journal article
A remarkable number of different numerical algorithms can be understood
and analyzed using the concepts of symmetric spaces and Lie triple systems, which
are well known in differential geometry from the study of spaces of constant curvature
and their tangents. This theory can be used to unify a range of different topics, such
as polar-type matrix decompositions, splitting methods for computation of the matrix
exponential, composition of selfadjoint numerical integrators and dynamical systems
with symmetries and reversing symmetries. The thread of this paper is the following:
involutive automorphisms on groups induce a factorization at a group level, and a
splitting at the algebra level. In this paper we will give an introduction to the mathematical
theory behind these constructions, and review recent results. Furthermore,
we present a new Yoshida-like technique, for self-adjoint numerical schemes, that
allows to increase the order of preservation of symmetries by two units. The proposed
techniques has the property that all the time-steps are positive.
Sat, 01 Mar 2014 00:00:00 GMThttp://hdl.handle.net/1956/90022014-03-01T00:00:00ZOn the Formulation of Mass, Momentum and Energy Conservation in the KdV Equationhttp://hdl.handle.net/1956/8968
On the Formulation of Mass, Momentum and Energy Conservation in the KdV Equation
Ali, Alfatih Mohammed A.; Kalisch, Henrik
Journal article
The Korteweg-de Vries (KdV) equation is widely recognized as a simple model
for unidirectional weakly nonlinear dispersive waves on the surface of a shallow body of
fluid. While solutions of the KdV equation describe the shape of the free surface, information
about the underlying fluid flow is encoded into the derivation of the equation, and the
present article focuses on the formulation of mass, momentum and energy balance laws in
the context of the KdV approximation. The densities and the associated fluxes appearing in
these balance laws are given in terms of the principal unknown variable η representing the
deflection of the free surface from rest position. The formulae are validated by comparison
with previous work on the steady KdV equation. In particular, the mass flux, total head and
momentum flux in the current context are compared to the quantities Q, R and S used in
the work of Benjamin and Lighthill (Proc. R. Soc. Lond. A 224:448–460, 1954) on cnoidal
waves and undular bores.
Wed, 01 Oct 2014 00:00:00 GMThttp://hdl.handle.net/1956/89682014-10-01T00:00:00ZElectrical conductivity of fractured media: A computational study of the self-consistent methodhttp://hdl.handle.net/1956/8923
Electrical conductivity of fractured media: A computational study of the self-consistent method
Sævik, Pål Næverlid; Berre, Inga; Jakobsen, Morten; Lien, Martha
Conference object
Effective medium theory can be used to link conductivity estimation
methods with prior knowledge about the distribution
of fractures in the investigated geological structure. In the literature,
little work has been presented on assessing the accuracy
of effective medium approximations for dense networks
of finite-sized fractures. We present here a systematic computational
study, comparing the conductivity predictions of
the popular self-consistent method with results from numerical
finite-element simulations. Our results show that the selfconsistent
method is accurate within acceptable error bounds
for a range of parameter values, in some cases even beyond
the percolation limit. We also compare the percolation thresholds
predicted by self-consistent theory with the thresholds obtained
by a numerical percolation algorithm. For the cases we
have studied, the percolation thresholds agree to a remarkable
degree.
Sun, 01 Jan 2012 00:00:00 GMThttp://hdl.handle.net/1956/89232012-01-01T00:00:00ZFinite volume hydromechanical simulation in porous mediahttp://hdl.handle.net/1956/8911
Finite volume hydromechanical simulation in porous media
Nordbotten, Jan Martin
Journal article
Cell-centered finite volume methods are prevailing in numerical simulation of flow in porous media. However, due to the lack of cell-centered finite volume methods for mechanics, coupled flow and deformation is usually treated either by coupled finite-volume-finite element discretizations, or within a finite element setting. The former approach is unfavorable as it introduces two separate grid structures, while the latter approach loses the advantages of finite volume methods for the flow equation. Recently, we proposed a cell-centered finite volume method for elasticity. Herein, we explore the applicability of this novel method to provide a compatible finite volume discretization for coupled hydromechanic flows in porous media. We detail in particular the issue of coupling terms, and show how this is naturally handled. Furthermore, we observe how the cell-centered finite volume framework naturally allows for modeling fractured and fracturing porous media through internal boundary conditions. We support the discussion with a set of numerical examples: the convergence properties of the coupled scheme are first investigated; second, we illustrate the practical applicability of the method both for fractured and heterogeneous media.
Tue, 27 May 2014 00:00:00 GMThttp://hdl.handle.net/1956/89112014-05-27T00:00:00ZControllability on Infinite-Dimensional Manifolds: A Chow-Rashevsky Theoremhttp://hdl.handle.net/1956/8886
Controllability on Infinite-Dimensional Manifolds: A Chow-Rashevsky Theorem
Khajeh Salehani, Mahdi; Markina, Irina
Journal article
One of the fundamental problems in control theory is that of controllability, the question of whether one can drive the system from one point to another with a given class of controls. A classical result in geometric control theory of finite-dimensional (nonlinear) systems is Chow–Rashevsky theorem that gives a sufficient condition for controllability on any connected manifold of finite dimension. In other words, the classical Chow–Rashevsky theorem, which is in fact a primary theorem in subriemannian geometry, gives a global connectivity property of a subriemannian manifold. In this paper, following the unified approach of Kriegl and Michor (The Convenient Setting of Global Analysis, Mathematical Surveys and Monographs, vol. 53, Am. Math. Soc., Providence, 1997) for a treatment of global analysis on a class of locally convex spaces known as convenient, we give a generalization of Chow–Rashevsky theorem for control systems in regular connected manifolds modelled on convenient (infinite-dimensional) locally convex spaces which are not necessarily normable. To indicate an application of our approach to the infinite-dimensional geometric control problems, we conclude the paper with a novel controllability result on the group of orientation-preserving diffeomorphisms of the unit circle.
Tue, 29 Apr 2014 00:00:00 GMThttp://hdl.handle.net/1956/88862014-04-29T00:00:00ZMethods and Tools for Analysis of Symmetric Cryptographic Primitiveshttp://hdl.handle.net/1956/8828
Methods and Tools for Analysis of Symmetric Cryptographic Primitives
Kazymyrov, Oleksandr
Doctoral thesis
<p>The development of modern cryptography is associated with the emergence of computing machines. Since specialized equipment for protection of sensitive information was initially implemented only in hardware, stream ciphers were widespread. Later, other areas of symmetric and asymmetric cryptography were established with the invention of general-purpose processors. In particular, such symmetric cryptographic primitives as block ciphers, message authentication codes (MACs), authenticated ciphers and others began to develop rapidly. Today various cryptographic algorithms are commonly used in everyday life to protect private data.</p>
<p>Design and analysis of advanced symmetric cryptographic primitives require a lot of time and resources. This is related to many factors, mainly to the cryptanalysis of prospective encryption algorithms under development. Every year new and modified attacks are published, leading to a rapid increase in the quantity of requirements and criteria imposed on cryptoprimitives.</p>
<p>Most of this thesis is devoted to analysis and improvement of cryptographic attacks and corresponding criteria for basic components. Almost all modern cryptoprimitives use nonlinear mappings for protection against advanced attacks. In connection with that a new method was proposed for the generation of random substitutions (S-boxes) with extreme cryptographic indicators that can be used in the next-generation ciphers to provide high and ultra-high security levels. In addition, several criteria imposed on S-boxes used in block ciphers were analyzed and their significance for block ciphers was proven. It is worth mentioning a practical method of testing two vectorial Boolean functions and a universal tool for checking properties of arbitrary binary nonlinear components presented in papers gathered in this thesis.</p>
<p>Another part of the thesis is dedicated to the cryptanalysis of hash functions as well as block and stream ciphers. To be more precise, an algebraic attack based on a binary decision diagram (BDD) was performed on the reduced Data Encryption Standard (DES), a scaled-down version of Advanced Encryption Standard (AES) and extended affine (EA) equivalence problem. Moreover, an algebraic approach was used to reconstruct an initial representation of the current Russian hash standard GOST 34.11-2012. Finally, a backward states tree method has been used to analyze stream ciphers based on the combination principle of linear and nonlinear feedback registers.</p>
Mon, 01 Dec 2014 00:00:00 GMThttp://hdl.handle.net/1956/88282014-12-01T00:00:00ZClassical and Stochastic Slit Löwner Evolutionhttp://hdl.handle.net/1956/8746
Classical and Stochastic Slit Löwner Evolution
Ivanov, Georgy
Doctoral thesis
Fri, 10 Oct 2014 00:00:00 GMThttp://hdl.handle.net/1956/87462014-10-10T00:00:00ZImproving the error rates of the Begg and Mazumdar test for publication bias in fixed effects meta-analysishttp://hdl.handle.net/1956/8611
Improving the error rates of the Begg and Mazumdar test for publication bias in fixed effects meta-analysis
Gjerdevik, Miriam; Heuch, Ivar
Journal article
<p>Background: The rank correlation test introduced by Begg and Mazumdar is extensively used in meta-analysis to test for publication bias in clinical and epidemiological studies. It is based on correlating the standardized treatment effect with the variance of the treatment effect using Kendall&rsquo;s tau as the measure of association. To our knowledge, the operational characteristics regarding the significance level of the test have not, however, been fully assessed.</p>
<p>Methods: We propose an alternative rank correlation test to improve the error rates of the original Begg and Mazumdar test. This test is based on the simulated distribution of the estimated measure of association, conditional on sampling variances. Furthermore, Spearman&rsquo;s rho is suggested as an alternative rank correlation coefficient. The attained level and power of the tests are studied by simulations of meta-analyses assuming the fixed effects model.</p>
<p>Results: The significance levels of the original Begg and Mazumdar test often deviate considerably from the nominal level, the null hypothesis being rejected too infrequently. It is proven mathematically that the assumptions for using the rank correlation test are not strictly satisfied. The pairs of variables fail to be independent, and there is a correlation between the standardized effect sizes and sampling variances under the null hypothesis of no publication bias. In the meta-analysis setting, the adverse consequences of a false negative test are more profound than the disadvantages of a false positive test. Our alternative test improves the error rates in fixed effects meta-analysis. Its significance level equals the nominal value, and the Type II error rate is reduced. In small data sets Spearman&rsquo;s rho should be preferred to Kendall&rsquo;s tau as the measure of association.</p>
<p>Conclusions: As the attained significance levels of the test introduced by Begg and Mazumdar often deviate greatly from the nominal level, modified rank correlation tests, improving the error rates, should be preferred when testing for publication bias assuming fixed effects meta-analysis.</p>
Mon, 22 Sep 2014 00:00:00 GMThttp://hdl.handle.net/1956/86112014-09-22T00:00:00ZA study in Univalencehttp://hdl.handle.net/1956/8566
A study in Univalence
Husebø, Anders Knarvik
Master thesis
In this master thesis we want to study the newly discovered homotopy type theory, and its
models within mathematics. We look at models in simplicial sets, simplicial symmetric monoids, and a new category which could be called multi pointed simplicial sets. We also describe dependent type theory from the informatical point of view, and some implications of it.
Mon, 02 Jun 2014 00:00:00 GMThttp://hdl.handle.net/1956/85662014-06-02T00:00:00ZPortfolio Optimization with PCC-GARCH-CVaR modelhttp://hdl.handle.net/1956/8555
Portfolio Optimization with PCC-GARCH-CVaR model
Xi, Linda Mon
Master thesis
This thesis investigates the Conditional Value-at-Risk (CVaR) portfolio optimization approach combined with a univariate GARCH model and pair-copula constructions (PCC) to determine the optimal asset allocation for a portfolio.
The methodology focuses on minimizing CVaR as the risk measure in replacement of variance used in the traditional optimization framework of Markowitz. GARCH model provides a tool for predicting and analyzing the time-varying volatility financial assets are exposed to, while copulas allow us to model the non-linear dependence structure and margins separately.
We compare the performance of the CVaR optimized portfolio with other investment strategies such as Constant-Mix and Buy-and-Hold. Although the selection of strategy depends on the investor risk profile, it is empirically shown that the proposed CVaR optimized portfolio outperforms the other two investment strategies based on the accumulated wealth in the long run.
Mon, 02 Jun 2014 00:00:00 GMThttp://hdl.handle.net/1956/85552014-06-02T00:00:00ZInvestigations of the Modified Navier-Stokes Equations in One Dimensionhttp://hdl.handle.net/1956/8554
Investigations of the Modified Navier-Stokes Equations in One Dimension
Sårheim, Inga Sofie
Master thesis
In this thesis we have investigated the Brenner-Navier-Stoke equations in one spatial dimension. These are a modified version of the Navier-Stokes equations, where the modification relates to a mass diffusion term. This modification would be significant for flows with high density gradients. In this case we will examine a shock wave problem in Argon. Both the original and the modified Navier-Stokes equations will be used to solve the conservation laws. We will study the effect of both entropy-stable and entropy-conservative schemes, in addition to several several different ways to model the diffusion parameters.
The solutions will be analyzed, and compared with experimental results.
Mon, 02 Jun 2014 00:00:00 GMThttp://hdl.handle.net/1956/85542014-06-02T00:00:00ZContinuous Max-Flow for Image Segmentation with Shape Priorshttp://hdl.handle.net/1956/8367
Continuous Max-Flow for Image Segmentation with Shape Priors
Kvile, Mari Aurlien
Master thesis
In this thesis we propose a stable method for image segmentation with shape priors. The original Chan-Vese intensity based segmentation model with regularisation term is extended to include shape prior information. We study shape priors which are pose invariant under the group of similarity transformations, that is under rotation, scaling and translation. In order to solve this problem robustly and effectively, an algorithm based on the theory of max-flow and min-cut is used in addition to a gradient descent procedure for updating the pose parameters. Comprehensive experiments are provided to demonstrate the behaviour of the proposed method on different images.
Mon, 02 Jun 2014 00:00:00 GMThttp://hdl.handle.net/1956/83672014-06-02T00:00:00ZFast Image Segmentation Using Variational Optimization Methods With Edge Detectorhttp://hdl.handle.net/1956/8366
Fast Image Segmentation Using Variational Optimization Methods With Edge Detector
Stanislaus, Mary Gerina
Master thesis
In this work, we apply techniques in variational optimization to image segmentation. We study three different segmentation models: one is based on the active contour method, the second is based on a piecewise constant level set method, and the last uses a continuous max-flow min-cut model. We obtain significantly better segmentation results in the first and the third model by including an experimental edge detector. The first model is a special case of the minimal partition problem, the second model uses discontinuities of piecewise constant level set functions to represent interfaces between the region of interest and the background, and the third model uses a spatially continuous max-flow min-cut framework which is a very efficient method to segment images. The first two models are non-convex and may contain many local solutions, but the last model is a convex optimization problem and therefore finds the global solution.
Mon, 02 Jun 2014 00:00:00 GMThttp://hdl.handle.net/1956/83662014-06-02T00:00:00ZExact and Superconvergent Solutions of the Multi-Point Flux Approximation O-method: Analysis and Numerical Testshttp://hdl.handle.net/1956/8353
Exact and Superconvergent Solutions of the Multi-Point Flux Approximation O-method: Analysis and Numerical Tests
Olderkjær, Daniel Stensrud
Master thesis
In this thesis we prove the multi-point flux approximation O-method (MPFA)
to yield exact potential and flux for the trigonometric potential functions
u(x,y)=sin(x)sin(y) and u(x,y)=cos(x)cos(y). This is done on uniform square grids
in a homogeneous medium with principal directions of the permeability aligned with
the grid directions when having periodic boundary conditions. Earlier theoretical
and numerical convergence articles suggests that these potential functions should
only yield second order convergence. Hence, our motivation for the analysis was to
gain new insight into the convergence of the method, as well as to develop
theoretical proofs for what seems as decent examples for testing implementation.
An extension of the result to uniform rectangular grids in an isotropic medium is also
briefly discussed, before we develop a numerical overview of the exactness phenomenon for
different types of boundary conditions. Lastly, an investigation of application of these
results to obtain exact potential and flux using the MPFA method for general potential
functions approximated by Fourier series was conducted.
Mon, 02 Jun 2014 00:00:00 GMThttp://hdl.handle.net/1956/83532014-06-02T00:00:00ZInvestigations of the Kaup-Boussinesq model equations for water waveshttp://hdl.handle.net/1956/8335
Investigations of the Kaup-Boussinesq model equations for water waves
Juliussen, Bjørn-Sverre Hjelle
Master thesis
The Kaup-Boussinesq system is a coupled system of nonlinear partial differential equations which has been derived as a model for surface waves in the context of the Boussinesq scaling, and it has also been derived for an internal wave system. In this thesis, modeling properties of the Kaup-Boussinesq water-wave model are under investigation. Differential balance laws for mass, momentum and energy are considered, and we present an exact differential balance for momentum. A Kaup-Boussinesq system describing long internal waves is investigated and compared with the Gardner equation. Finally, a spectral method for the numerical discretization of the Kaup-Boussinesq system for surface waves is put forward, and shown to converge and be stable.
Mon, 02 Jun 2014 00:00:00 GMThttp://hdl.handle.net/1956/83352014-06-02T00:00:00ZRadielle basisfunksjoner som adaptiv kollokasjonsmetodehttp://hdl.handle.net/1956/8333
Radielle basisfunksjoner som adaptiv kollokasjonsmetode
Nicolajsen, Tomas
Master thesis
Hensikten med oppgaven er å undersøke en adaptiv kollokasjonsmetode basert
på radielle basisfunksjoner. Målet er å finne ut om man med enkle kriterier
for senter- og formparameterfordelingen kan oppnå en fordeling som
gjenspeiler egenskaper ved problemet, og om man med det kan oppnå en mer
effektiv og stabil kollokasjonsmetode.
Sun, 01 Jun 2014 00:00:00 GMThttp://hdl.handle.net/1956/83332014-06-01T00:00:00ZBegrepsforståelse i matematikkfagethttp://hdl.handle.net/1956/8332
Begrepsforståelse i matematikkfaget
Vinnes, Eivind Ole
Master thesis
Intensjonen med denne oppgaven er å kunne gi en vurdering av hvilke matematiske basiskunnskaper elevene som begynner på videregående har og gi noen forslag til hva som kan gjøres for å forbedre undervisningen
Fri, 30 May 2014 00:00:00 GMThttp://hdl.handle.net/1956/83322014-05-30T00:00:00ZRekneark på ungdomstrinnethttp://hdl.handle.net/1956/8331
Rekneark på ungdomstrinnet
Johansson, Kari
Master thesis
Problemstillinga i oppgåva var korleis opplever elevane bruk av rekneark i matematikk på ungdomstrinnet. Gjennom eitt år vart ulike reknearkoppgåver prøvd ut i 9./10. klasse. I etterkant henta eg inn data ved hjelp av spørjeskjema, opptaksoppgåver av to og to elevar som løyste utvalde oppgåver og eit gruppeintervju. Eg la vekt på kva fordelar og problem elevane opplevde, og kva ulike typar oppgåver som er relevante for elevane.
Fri, 06 Jun 2014 00:00:00 GMThttp://hdl.handle.net/1956/83312014-06-06T00:00:00ZHybrids between common and Antarctic minke whales are fertile and can back-crosshttp://hdl.handle.net/1956/8118
Hybrids between common and Antarctic minke whales are fertile and can back-cross
Glover, Kevin; Kanda, Naohisa; Haug, Tore; Pastene, Luis A.; Øien, Nils Inge; Seliussen, Bjørghild Breistein; Sørvik, Anne Grete Eide; Skaug, Hans J.
Journal article
<p>Background:
Minke whales are separated into two genetically distinct species: the Antarctic minke whale found in
the southern hemisphere, and the common minke whale which is cosmopolitan. The common minke whale is
further divided into three allopatric sub-species found in the North Pacific, southern hemisphere, and the North
Atlantic. Here, we aimed to identify the genetic ancestry of a pregnant female minke whale captured in the North
Atlantic in 2010, and her fetus, using data from the mtDNA control region, 11 microsatellite loci and a sex
determining marker.</p><p>Results:
All statistical parameters demonstrated that the mother was a hybrid displaying maternal and paternal
contribution from North Atlantic common and Antarctic minke whales respectively. Her female fetus displayed
greater genetic similarity to North Atlantic common minke whales than herself, strongly suggesting that the hybrid
mother had paired with a North Atlantic common minke whale.</p><p>Conclusion:
This study clearly demonstrates, for the first time, that hybrids between minke whale species may be
fertile, and that they can back-cross. Whether contact between these species represents a contemporary event
linked with documented recent changes in the Antarctic ecosystem, or has occurred at a low frequency over many
years, remains open.</p>
Mon, 15 Apr 2013 00:00:00 GMThttp://hdl.handle.net/1956/81182013-04-15T00:00:00ZReversible Jump Markov Chain Monte Carlo: Some Theory and Applicationshttp://hdl.handle.net/1956/8023
Reversible Jump Markov Chain Monte Carlo: Some Theory and Applications
Lyyjynen, Hannu Sakari
Master thesis
The history of MCMC, theories of Bayesian thinking
and model choice, the Accept-
Reject-algorithm, Markov chains, the Metropolis-
Hastings-algorithm and the reversible
jump MCMC are explained. Then the reversible jump
MCMC as change-point analysis
is applied to the coal mine disaster example,
familiar from [Green, 1995], and to the
examples of counting terrorism attacks (worldwide,
in Iraq and in Afghanistan). The
novel part is estimating the change points of the
hazard rate of terrorism attacks in
Afghanistan during the last 35 years.
Mon, 10 Mar 2014 00:00:00 GMThttp://hdl.handle.net/1956/80232014-03-10T00:00:00ZGeological Storage of CO2: Sensitivity and Risk Analysishttp://hdl.handle.net/1956/7913
Geological Storage of CO2: Sensitivity and Risk Analysis
Ashraf, Meisam
Doctoral thesis
<p>Geological CO2 storage has the potential to be a key technology for prevention of industrial CO2 emission
into the atmosphere. A successful storage operation requires safe geological structures with large
storage capacity. The practicality of the technology is challenged by various operational concerns,
ranging from site selection to long-term monitoring of the injected CO2. The research in this report addresses
the value of using sophisticated geological modeling to help in predicting storage performance.</p><p>In the first part, we investigate the significance of assessing the geological uncertainty and its consequences
in site selection and the early stages of storage operations. This includes the injection period
and the early migration time of the injected CO2 plume. The extensive set of realistic geological realizations
used in the analysis makes the key part of this research. Heterogeneity is modelled using the
most influential geological parameters in a shallow-marine system, including aggradation angle, levels
of barriers in the system, faults, lobosity, and progradation direction.</p><p>A typical injection scenario is simulated over 160 realizations and major flow responses are defined
to measure the success of the early stages of CO2 storage operations. These responses include the
volume of trapped CO2 by capillarity, dynamics of the plume in the medium, pressure responses, and
the risk of leakage through a failure in the sealing cap-rock. The impact of geological uncertainty
on these responses is investigated by comparing all cases for their performance. The results show
large variations in the responses due to changing geological parameters. Among the main influential
parameters are aggradation angle, progradation direction, and faults in the medium.</p><p>A sophisticated geological uncertainty study requires a large number of detailed simulations that
are time-consuming and computationally costly. The second part of the research introduces a workflow
that employs an approximating response surface method called arbitrary polynomial chaos (aPC). The
aPC is fast and sophisticated enough to be used practically in the process of sensitivity analysis and
uncertainty and risk assessment. We demonstrate the workflow by combining the aPC with a global
sensitivity analysis technique, the Sobol indices, which is a variance-based method proven to be practical
for complicated physical problems. Probabilistic uncertainty analysis is performed by applying
the Monte Carlo process using the aPC. The results show that the aPC can be used successfully in an
extensive geological uncertainty study.</p>
Thu, 10 Apr 2014 00:00:00 GMThttp://hdl.handle.net/1956/79132014-04-10T00:00:00ZStatistiske metoder for alder-periode-kohort-analyser. En sammenligning av nyere metoder med konvensjonelle generaliserte lineære modellerhttp://hdl.handle.net/1956/7795
Statistiske metoder for alder-periode-kohort-analyser. En sammenligning av nyere metoder med konvensjonelle generaliserte lineære modeller
Askeland, Olaug Margrete
Master thesis
Masteroppgaven tar for seg statistiske metoder for alder-periode-kohort-analyser (APC-analyser). For en gitt respons forsøker APC-analyser å separere den påvirkningen som skyldes alder, fra den påvirkningen som er assosiert med tidsperiode, og den påvirkning som er assosiert med fødselstidspunkt. Den velkjente sammenhengen mellom de tre faktorene, periode - alder = kohort, gjør parameterestimeringen vanskelig, og et generelt dilemma ved APC-analyser er problemstillingen med å separere de simultane effektene. Det har vært foreslått mange løsninger for dette identifikasjonsproblemet og i denne oppgaven sammenlignes nyere metoder mot konvensjonelle generaliserte lineære modeller. Sammenligningene blir gjort med simuleringsanalyser. Det er spesielt tatt utgangspunkt i en nyere metode, Intrinsic Estimator (IE) og sammenlignet denne med en mye benyttet metode, Constrained Generalized Linear Models estimator (CGLIM). CGLIM-metoden er avhengig av at det innføres en betingelse for koeffisientene, og problemet med denne metoden er at den avhenger av forhåndsinformasjon om dataene for å sette disse betingelsene. Estimatene til koeffisientene er sensitiv for valget av denne betingelsen. IE-metoden forsøker å oppnå modellidentifikasjon med minimale antagelser. Videre er det også foretatt en sammenligning med metoder som baserer seg på førsteordensdifferanser og andreordensdifferanser og metoder som inkluderer drift. Ved analyse av enkelte datasett trenger man ikke benytte den fulle APC-modellen, men det holder å benytte metoder som kun inkluderer en eller to av faktorene. Ulike goodness-of-fit-mål er da nyttig for å vurdere om en modell tilpasser et datasett godt nok. IE-metoden er også sammenlignet med slike metoder. En nyere metode som baserer seg på Partial Least Squares for å estimere de simultane effektene til alder, periode og kohort introduseres også.
IE-metoden har vist seg å være en nyttig tilnærming for identifikasjon og estimering i APC-modellen og produserer forventningsrette og effisiente estimater. Den er et sikrere valg når en ikke vet noe om de dataene en skal analysere, for i slike tilfeller kan valg av en vilkårlig betingelse i de andre metodene i verste fall gi estimater som er langt unna sannheten.
Wed, 20 Nov 2013 00:00:00 GMThttp://hdl.handle.net/1956/77952013-11-20T00:00:00ZOptimization of CO2 Geological Storage Costhttp://hdl.handle.net/1956/7704
Optimization of CO2 Geological Storage Cost
Zhu, Sha
Master thesis
Carbon dioxide capture and storage (CCS) is a promising strategy to battle the climate change by injecting large-scale of carbon dioxide back to underground formations and storing the carbon dioxide possibly permanently. It is an existing technology but for climate and economic concern it is still a relevantly new concept. We are interested in studying the cost of CCS, in particular the cost of CO2 geological storage, and optimize the cost, since the high cost of CCS is a big hurdle for industry to deploy this technology.
Sun, 01 Dec 2013 00:00:00 GMThttp://hdl.handle.net/1956/77042013-12-01T00:00:00ZModeling and simulation of concrete carbonation in 1-d using two-point flux approximationhttp://hdl.handle.net/1956/7689
Modeling and simulation of concrete carbonation in 1-d using two-point flux approximation
Røe, Tineke
Master thesis
In this master's thesis we will model concrete carbonation using mass conservation equations and Darcy's law. We then get a set of coupled partial differential equations. We also have a ordinary differential equation modeling porosity change. These equations are discretized using
two-point flux approximation for 1-d in space, and Euler implicit in time. We have a nonlinear pressure equation which is linearized using Newton method. We run simulations, and the results are compared to a constructed analytical solution and to examples in the literature.
Thu, 21 Nov 2013 00:00:00 GMThttp://hdl.handle.net/1956/76892013-11-21T00:00:00ZNegativ binomisk regresjon med modifiserte sannsynligheter for nullobservasjoner; ZINB og ZANBhttp://hdl.handle.net/1956/7686
Negativ binomisk regresjon med modifiserte sannsynligheter for nullobservasjoner; ZINB og ZANB
Optun, Mette
Master thesis
Regresjonsmodeller av Poisson- og negativ binomisk fordeling blir ofte brukt til å utføre regresjonsanalyse på datasett innen medisin, biologi, økonomi og mange andre fagfelt. Det oppstår ofte tilfeller der andelen av verdien null i datasettet ikke samstemmer med den som er forventet når det antas at observasjonene enten er negativ binomisk (NB)- eller Poissonfordelt. Et eksempel på slike kan være antall innrapporterte skademeldinger et år fra en gruppe forsikringstakere, som grunnet egenandel vil inneholde flere observasjoner av null enn forventet. Antall dager en pasient er innlagt på sykehus, der kun innlagte pasienter er tatt med i beregningen, er et annet eksempel på at forventet antall nullobservasjoner ikke vil være lik forventet antall fra NB- eller Poisson fordelte data.
Denne oppgaven vil ta for seg to modeller som er laget for å tilpasse datasett som er antatt negativ binomisk fordelt, men som inneholder mange nullobservasjoner i tillegg.
Før en introduksjon av disse modellene, er det valgt å inkludere en del teori om negativ binomisk fordelinger og regresjonsmodeller som tar utgangspunkt i disse. Dette vil gi et godt grunnlag for å forstå oppbyggingen i fordelingene bak, og til modellene ZINB og ZANB, når disse deretter vil bli grundig forklart. Til slutt vil det undersøkes om valg av de to modellene er avgjørende for resultat etter tilpassing av ulike datasett, og hvilke konsekvenser bruk av enten ZINB eller ZANB som regresjonsmodell når den andre er den korrekte kan gi. I tillegg vil det vurderes om det generelt er mulig å se hvilken modell som er mest riktig ut ifra egenskaper i det observerte datasettet.
Wed, 06 Nov 2013 00:00:00 GMThttp://hdl.handle.net/1956/76862013-11-06T00:00:00ZVolume preserving numerical integrators for ordinary differential equationshttp://hdl.handle.net/1956/7563
Volume preserving numerical integrators for ordinary differential equations
Xue, Huiyan
Doctoral thesis
Fri, 08 Nov 2013 00:00:00 GMThttp://hdl.handle.net/1956/75632013-11-08T00:00:00ZGenerating function and volume preserving mappingshttp://hdl.handle.net/1956/7562
Generating function and volume preserving mappings
Xue, Huiyan; Zanna, Antonella
Peer reviewed
In this paper, we study generating forms and generating functions
for volume preserving mappings in Rn. We derive some parametric classes
of volume preserving numerical schemes for divergence free vector fields. In
passing, by extension of the Poincar´e generating function and a change of
variables, we obtained symplectic equivalent of the theta-method for differential
equations, which includes the implicit midpoint rule and symplectic Euler A
and B methods as special cases.
Sat, 01 Mar 2014 00:00:00 GMThttp://hdl.handle.net/1956/75622014-03-01T00:00:00ZHigh order volume preserving integrators for three kinds of divergence-free vector fields via commutatorshttp://hdl.handle.net/1956/7561
High order volume preserving integrators for three kinds of divergence-free vector fields via commutators
Xue, Huiyan
Journal article
In this paper, we focus on construction of high order volume preserving integrators
for divergence-free vector fields of the monomial basis, exponential basis and tensor
product of the monomial and exponential basis. We first prove that the commutators
of elementary divergence-free vector fields (EDFVF) of these three kinds are
still divergence-free vector fields of the same kind. For EDFVFs of these three kinds,
we construct high order volume preserving integrators using the multi-commutators.
Moreover, we consider ordering of EDFVFs and their commutators to reduce the error
of the schemes, showing by numerical tests that the strategies in [8] work well.
Tue, 01 Jan 2013 00:00:00 GMThttp://hdl.handle.net/1956/75612013-01-01T00:00:00ZDetecting Periodic Elements in Higher Topological Hochschild Homologyhttp://hdl.handle.net/1956/7560
Detecting Periodic Elements in Higher Topological Hochschild Homology
Veen, Torleif
Doctoral thesis
Tue, 20 Aug 2013 00:00:00 GMThttp://hdl.handle.net/1956/75602013-08-20T00:00:00ZImage processing, filtering and segmentation of data-sets for reservoir simulationhttp://hdl.handle.net/1956/7543
Image processing, filtering and segmentation of data-sets for reservoir simulation
Gurholt, Tiril Pedersen
Doctoral thesis
Paper A: 2-D Visualisation of Unstable Waterflood and Polymer Flood for Displacement
of Heavy Oil. Full-text not available in BORA. The published version is available at: <a href="http://dx.doi.org/10.2118/154292-ms" target="blank">http://dx.doi.org/10.2118/154292-ms</a>; Paper B: Determination of Connectivity in Vuggy Carbonate Rock Using Image
Segmentation Techniques. Full-text not available in BORA.; Paper C: Pore space characterization of vuggy carbonate rocks: A comparative
study of the performance of various image segmentation techniques. Full-text not available in BORA.; Paper D: 3D Multiphase Piecewise Constant Level Set Method Based on Graph
Cut Minimization. Full-text not available in BORA.
Fri, 11 Oct 2013 00:00:00 GMThttp://hdl.handle.net/1956/75432013-10-11T00:00:00ZModelling Fluid Flow and Heat Transport in Fractured Porous Mediahttp://hdl.handle.net/1956/7509
Modelling Fluid Flow and Heat Transport in Fractured Porous Media
Lampe, Victor
Master thesis
Flow in porous media is an important and well-
researched topic in both academia and the
industry. More accurate mathematical models are
constantly sought after.
Fractures in a porous medium can be an important
contributor
to the overall dynamics of a porous system, and
several conceptual models for fractured porous
media has been proposed
over the years.
We investigate a selection of these models, by
starting
with the basics of non-fractured porous media.
We discuss the effects of fractures, concerning
both single-phase fluid flow and heat transport,
and review
a few different approaches for the modelling of
fractured porous media.
Three different models, and equivalent continuum
model, a dual continuum model and a discrete
fracture model, are compared
through analysis and numerical experiments.
Our simulations show that the equivalent continuum
model is unreliable and coming up short to the
other two, from which we obtain comparable
results.
Tue, 27 Aug 2013 00:00:00 GMThttp://hdl.handle.net/1956/75092013-08-27T00:00:00ZAuxiliary variables for 3D multiscale simulations in heterogeneous porous mediahttp://hdl.handle.net/1956/7429
Auxiliary variables for 3D multiscale simulations in heterogeneous porous media
Sandvin, Andreas; Keilegavlen, Eirik; Nordbotten, Jan Martin
Journal article
<p>The multiscale control-volume methods for solving problems involving flow in porous media have gained
much interest during the last decade. Recasting these methods in an algebraic framework allows one to
consider them as preconditioners for iterative solvers. Despite intense research on the 2D formulation,
few results have been shown for 3D, where indeed the performance of multiscale methods deteriorates. The
interpretation of multiscale methods as vertex based domain decomposition methods, which are non-scalable
for 3D domain decomposition problems, allows us to understand this loss of performance.</p><p>We propose a generalized framework based on auxiliary variables on the coarse scale. These are enrichments
the coarse scale, which can be selected to improve the interpolation onto the fine scale. Where the
existing coarse scale basis functions are designed to capture local sub-scale heterogeneities, the auxiliary variables
are aimed at better capturing non-local effects resulting from non-linear behavior of the pressure field.
The auxiliary coarse nodes fits into the framework of mass-conservative domain-decomposition (MCDD)
preconditioners, allowing us to construct, as special cases, both the traditional (vertex based) multiscale
methods as well as their wire basket generalization.</p>
Mon, 01 Apr 2013 00:00:00 GMThttp://hdl.handle.net/1956/74292013-04-01T00:00:00ZA unified multilevel framework of upscaling and domain decompositionhttp://hdl.handle.net/1956/7417
A unified multilevel framework of upscaling and domain decomposition
Sandvin, Andreas; Nordbotten, Jan Martin; Aavatsmark, Ivar
Conference object
We consider multiscale preconditioners for a class of mass-conservative
domain-decomposition (MCDD) methods. For the application of reservoir simulation,
we need to solve large linear systems, arising from finite-volume discretisations of elliptic
PDEs with highly variable coefficients. We introduce an algebraic framework, based on
probing, for constructing mass-conservative operators on a multiple of coarse scales. These
operators may further be applied as coarse spaces for additive Schwarz preconditioners.
By applying different local approximations to the Schur complement system based on a
careful choice of probing vectors, we show how the MCDD preconditioners can be both
efficient preconditioners for iterative methods or accurate upscaling techniques for the
heterogeneous elliptic problem. Our results show that the probing technique yield better
approximation properties compared with the reduced boundary condition commonly
applied with multiscale methods.
Presented at CMWR 2010 - XVIII International Conference on Computational Methods in Water Resources, June 21-24, 2010, Barcelona, Spain
Fri, 01 Jan 2010 00:00:00 GMThttp://hdl.handle.net/1956/74172010-01-01T00:00:00ZRobust Multiscale Control-volume Methods for Reservoir Simulationhttp://hdl.handle.net/1956/7416
Robust Multiscale Control-volume Methods for Reservoir Simulation
Sandvin, Andreas
Doctoral thesis
Mon, 30 Apr 2012 00:00:00 GMThttp://hdl.handle.net/1956/74162012-04-30T00:00:00ZMass Conservative Domain Decomposition for Porous Media Flowhttp://hdl.handle.net/1956/7415
Mass Conservative Domain Decomposition for Porous Media Flow
Nordbotten, Jan Martin; Keilegavlen, Eirik; Sandvin, Andreas
Chapter
Wed, 28 Mar 2012 00:00:00 GMThttp://hdl.handle.net/1956/74152012-03-28T00:00:00ZEnergieffektivitet i grunne geotermiske systemer. Modellering og analyse av systemet på Ljan skole.http://hdl.handle.net/1956/7109
Energieffektivitet i grunne geotermiske systemer. Modellering og analyse av systemet på Ljan skole.
Straalberg, Eirik Ask
Master thesis
Grunnvarme er en stadig viktigere og mer anvendt energikilde i en verden med et økende energibehov, og det er i den forbindelse viktig å utforme systemer som kan utnytte energien på en effektiv måte. Denne oppgaven tar for seg det grunne og lukkede geotermiske anlegget på Ljan skole i Oslo, der en geotermisk varmepumpe forsynes med varme fra en geotermisk brønnpark og en asfaltert bakkesolfanger. Et hovedmål med denne oppgaven er å vurdere energektiviteten til dette systemet. Dette gjøres ved hjelp av simuleringsverktøyet TRNSYS, der det lages systemmodeller av anlegget. Spesielt studeres borehullsbasert lagring av solvarme, og det vurderes om dette er en energieffektiv løsning. For verifisering av modeller og resulater sammenlignes enkelte simuleringsresulater med driftsdata fra Ljan skole.
Det konkluderes med at systemet på Ljan skole er energieffektivt, og beregningene tyder på at det geotermiske systemet gir nesten halverte fyringskostnader sammenlignet med et elektrisk eller oljefyrbasert oppvarmingsystem. Systemet ser imidlertid ikke ut til å være tjent med borehullsbasert energilagring. Simuleringer uten energilagring gir enda høyere energieffektivitet enn simuleringer med energilagring, også på lang sikt.
Mon, 17 Jun 2013 00:00:00 GMThttp://hdl.handle.net/1956/71092013-06-17T00:00:00ZSyntetisk CDO: iTraxx-prising ved bruk av en Normal Invers Gaussisk Copulahttp://hdl.handle.net/1956/7076
Syntetisk CDO: iTraxx-prising ved bruk av en Normal Invers Gaussisk Copula
Nordanå, Kjetil Sørlien
Master thesis
I denne oppgaven skal vi se at det er et stort
marked for kredittderivater etter finanskrisen,
men da i en vridning mot syntetiske varianter. Men
framtidsutsiktene er usikre, da disse produktene
er i en
særstilling når det gjelder nye reguleringer.
Likevel fører kredittderivater med seg store
muligheter for å
flytte på risiko, slik at deltakere i markedet kan
bli avlastet endel i sine porteføljer. For
at markedsdeltakere som enten allerede er i en
eksisterende posisjon, eller vurderer
investeringer
på slike derivater, lettere kan gjøre opp
risikovurderinger er det viktig med modeller som
kan
reprodusere markedspriser med små avvik på en
tidseffektiv måte.
Gjennom oppgaven blir derfor tre modeller
presentert og prøvd ut: Gaussisk-LHP, Student
t-copula og NIG-LHP på den transjerte iTraxx
Europe-indeksen. Vi vil da se at det kun er
sistnevnte som på en effektiv måte reproduserer
markedspriser tilfredsstillende hvorpå den alltid
priser junior messanintransjen korrekt. Årsakene
til dette er de andre modellenens manglende
evne til å produsere korrelasjonssmilet observert
i markedet. I tillegg har dem kun en parameter,
korrelasjonen, som ikke er nok for å produsere en
slik kompleks struktur som syntetisk CDO
er. NIG har flere (intuitive) parametere i tillegg
til korrelasjonen og dermed føyer seg lettere til
markedspriser. I tillegg er NIG-modellen vi
presenterer semi-analytisk slik at den bruker kort
tid på å beregne. Gjennom oppgaven skal vi
utforske dens evne til å reprodusere markedspriser
for flere forskjellige tidsperioder, hvor vi
betraktet perioden under finanskrisen som svært
vanskelig, grunnet den lave likviditeten i
markedet. Med den pågående gjeldskrisen i de
europeiske landene, kan vi vente oss en stadig
ettersprøsel etter kredittderivater på europeiske
selskaper, slik at det er spennende å følge
utviklingen framover.
Thu, 04 Apr 2013 00:00:00 GMThttp://hdl.handle.net/1956/70762013-04-04T00:00:00ZSquare-free modules and ideals: Brill–Noether theory, polarizations, and deformations.http://hdl.handle.net/1956/6998
Square-free modules and ideals: Brill–Noether theory, polarizations, and deformations.
Lohne, Henning
Doctoral thesis
Fri, 24 May 2013 00:00:00 GMThttp://hdl.handle.net/1956/69982013-05-24T00:00:00ZBrill–Noether theory of squarefree modules supported on a graphhttp://hdl.handle.net/1956/6994
Brill–Noether theory of squarefree modules supported on a graph
Fløystad, Gunnar; Lohne, Henning
Journal article; Peer reviewed
We investigate the analogy between squarefree Cohen-
Macaulay modules supported on a graph and line bundles on a
curve. We prove a Riemann–Roch theorem, we study the Jacobian
and gonality of a graph, and we prove Clifford’s theorem.
Wed, 01 May 2013 00:00:00 GMThttp://hdl.handle.net/1956/69942013-05-01T00:00:00ZAnalysis of multi-beam sonar echos of herring schools by means of simulationhttp://hdl.handle.net/1956/6967
Analysis of multi-beam sonar echos of herring schools by means of simulation
Holmin, Arne Johannes
Doctoral thesis
<p>The synchronized behavior of large schools of fish can be a fascinating sight and has
caught the attention of researchers for decades. Schools of thousands, or even millions of
fish can seemingly function as a single entity, by mechanisms that are still not entirely
understood. Models describing the individual behavior (individual based models) have
been shown to predict certain schooling features such as predator avoidance, whereas
experiments with fish schools in tanks have revealed rules governing the interaction between
neighboring fish. However, the step from an individual based model or a school
of a limited number of fish in a tank experiment, to large free swimming schools in the
ocean, may include challenges with regards to the observation techniques and to conditions
of the environment that are not necessarily reproduced in the individual based
model or in the tank.</p> <p>The latest technological advances in underwater acoustic observation has introduced
the potential of observing schools of fish of size up to a few hundred meters by threedimensional
images generated by multi-beam sonars. The Simrad MS70 multi-beam
sonar provides true three-dimensional images at a temporal resolution down to approximately
one image per second, enabling observations of the dynamic behavior of large
fish schools in situ, at a spatial and temporal resolution that has not been previously
available. In this thesis, data from the MS70 sonar are analyzed by means of simulation,
and steps are taken towards establishing a link between the modeled behavior of an individual
and the observed behavior of real fish schools.</p> <p>The principal analytical tool utilized in the thesis is a simulation model developed by the
author (and co-authors), which simulates observations from multi-beam sonars based on
the positions, orientations, and acoustical properties of arbitrary groups of individual
fish, and the configuration and acoustical properties of the sonar and the environment.
The framework of the simulation model is presented in the first of three papers in the
thesis, along with examples of its use as implemented for the EK60 multi-frequency
echosounder, the ME70 multi-beam echosounder, and the MS70 multi-beam sonar, all
manufactured by Simrad. The simulation experiments shown in the first paper illustrate
for example the potential of the MS70 sonar to provide information about the behavior
of fish schools. Specifically, in one of the experiments, a herring school with original
mean heading perpendicular to the central sonar beams is modified to represent eight
different idealized orientation scenarios, obtained by rotating the fish in specific sections
of the school by 90° towards the sonar. This produced reduced backscatter in the sections
of the school which were rotated, due to the directionality of the scattering from
herring at the acoustic frequencies of the sonar, showing the potential for falsely interpreting
orientation changes as fish density changes, but also the potential for extracting information about the local orientation distribution of the school.</p> <p>In the second paper, the parameters and stochastic properties of the noise (defined
as all contributions to the received acoustic intensity not backscattered from targets)
present in data from the MS70 sonar and EK60 echosounder are estimated from passive
recording sequences, motivated by the following three potential uses: (1) subtraction
of noise from real data, (2) simulation of noise in synthetic data, and (3) to provide a
basis for the development of methods for segmentation of MS70 data of fish schools (applied
in the third paper). Particularly, a simulation experiment from the first paper is
repeated with addition of noise, in which the polarization (degree of alignment of the
individual fish) of a real school which had been circumnavigated by the vessel for several
rounds, was estimated by matching the total backscatter of the real school and the
total backscatter of simulated schools with a variety of polarizations and packing densities.
The experiment illustrates the effect of noise on the estimated polarization, and
the potential to infer packing density of the school from the simulation experiment.</p> <p>The estimated noise from the second paper is utilized in the third and final paper, in
the development and testing of a new segmentation method for multi-beam sonar data,
which is compared to an existing segmentation method implemented in the post processing
system LSSS (Large Scale Survey System). The new method applies a Bayesian
approach, where the cumulative distribution function (CDF) of the packing density of
omnidirectional targets (scattering equally in all directions) in a voxel is estimated based
on the observed data, the estimated noise, and a prior probability distribution of the
signal. The CDF is evaluated at a specified lower schooling threshold, and the resulting
probabilities that the packing density is below the schooling threshold is smoothed by a
Gaussian kernel in the logarithmic domain (implying products of the probabilities in the
neighborhood around the voxel). Voxels for which the smoothed probability is below a
segmentation threshold, are identified to contain the school. The two segmentation methods
are tested on simulated data of 240 herring schools of various shapes, sizes, packing
densities, and depths, and compared with ground truth segmentation data generated
from the fish positions used as input to the simulation model to identify recommended
parameter settings for both methods, and to determine differences in the performance
between the methods. The new method is shown to produce estimates of the school extent,
total volume, total target strength (total echo), and mean volume backscattering
strength (mean echo density) which are generally closer to the corresponding theoretical
values estimated from the ground truth segmentation data.</p>
Fri, 31 May 2013 00:00:00 GMThttp://hdl.handle.net/1956/69672013-05-31T00:00:00ZSimulations of multi-beam sonar echos from schooling individual fish in a quiet environmenthttp://hdl.handle.net/1956/6958
Simulations of multi-beam sonar echos from schooling individual fish in a quiet environment
Holmin, Arne Johannes; Handegard, Nils Olav; Korneliussen, Rolf J.; Tjøstheim, Dag
Journal article; Peer reviewed
A model is developed and demonstrated for simulating echosounder and sonar observations of fish
schools with specified shapes and composed of individuals having specified target strengths and
behaviors. The model emulates the performances of actual multi-frequency echosounders and
multi-beam echosounders and sonars and generates synthetic echograms of fish schools that can be
compared with real echograms. The model enables acoustic observations of large in situ fish
schools to be evaluated in terms of individual and aggregated fish behaviors. It also facilitates
analyses of the sensitivity of fish biomass estimates to different target strength models and their
parameterizations. To demonstrate how this tool may facilitate objective interpretations of
acoustically estimated fish biomass and behavior, simulated echograms of fish with different spatial
and orientation distributions are compared with real echograms of herring collected with a multibeam
sonar aboard the research vessel “G.O. Sars.” Results highlight the important effects of
fish-backscatter directivity, particularly when sensing with small acoustic wavelengths relative to
the fish length. Results also show that directivity is both a potential obstacle to estimating fish
biomass accurately and a potential source of information about fish behavior.
Sat, 01 Dec 2012 00:00:00 GMThttp://hdl.handle.net/1956/69582012-12-01T00:00:00ZComputer-aided proofs and algorithms in analysishttp://hdl.handle.net/1956/6817
Computer-aided proofs and algorithms in analysis
Bartha, Ferenc A.
Doctoral thesis
<p>The computational power has increased dramatically since the appearance of the first
computers, making them a vital tool in the analysis of dynamical systems. We present
further applications of those two basic ideas, namely interval arithmetic and automatic
differentiation, that address the question of the reliability of the results and the difficulty
of calculating derivatives.</p>
<p>In general, the result of a numerical calculation will be influenced by errors, since
the set of the numbers represented by the machine is finite. This will inevitably lead
to round-off and truncation errors. This should not be considered as a problem, but
rather as the true nature of numerics. The notorious examples like evaluating 333.75y6+
x2(11x2y2−y6−121y4−2)+5.5y8+x/(2y) at (x, y) = (77617,33096) or plotting the
polynomial t6 −6t5 +15t4 −20t3 +15t2 −6t +1 in a small neighborhood of 1, still
result in unexpected outcomes, if one is unaware of the potential risks of the floating
point computations. We mention the failure of a Patriot missile on February 25, 1991
or the explosion of the unmanned space rocket Ariane 5 on June 4, 1996 as practical
examples of these potential risks becoming real.</p>
<p>Therefore in mathematical proofs, where the beauty of the argument is its unquestionable
truth itself, the usage of computers must be handled with extreme care. One
technique, that is used to overcome these problems and make our computations rigorous,
is called interval arithmetic.</p>
<p>To calculate derivatives of a given function is often considered to be a hard problem,
since in general with increasing the order or the dimension, the complexity of the formula
of the derivative grows exponentially. The observation, that we do not need these
formulae in general, but only certain values of the derivatives, is crucial to understand
why automatic differentiation is so useful.</p>
<p>The structure of the thesis is as follows. In Part I we give an introduction to the
methods used in our papers. In Chapter 1 we get acquainted with the basic techniques,
interval arithmetic, interval analysis, floating point computations and automatic differentiation.
Chapter 2 gives an overview of the interaction between dynamical systems and different representations of the data. In Chapter 3 we take on the basic concept of automatic differentiation seen before, and present a method by Griewank et al. [17] to compute higher order derivatives of multivariate functions that will be used in Paper A. We go through the theory of graph representations in Chapter 4 by following the steps of Hohmann and Dellnitz [12] and Galias [15]. This theory may be used in qualitative analysis of maps. We give two applications in Paper B and Paper C. In addition, we
give the proof of correctness of the algorithm for enclosing non-wandering points in Paper B. In Chapter 5 we introduce the reader to the method of self-consistent bounds by
Zgliczy´nski and Mischaikow [44] and Zgliczy´nski [40, 42, 43] that may be used to analyze
a certain class of dissipative partial differential equations. An application of this
concept to a destabilized Kuramoto-Sivashinsky equation is given in Paper D. Chapter 6
gives a short overview of the results of the included papers.
Part II is the main scientific contribution of this thesis, consisting of the formerly
mentioned four papers.</p>
Fri, 14 Jun 2013 00:00:00 GMThttp://hdl.handle.net/1956/68172013-06-14T00:00:00ZIll Posedness Results for Generalized Water Wave Modelshttp://hdl.handle.net/1956/6802
Ill Posedness Results for Generalized Water Wave Models
Teyekpiti, Vincent Tetteh
Master thesis
In the first part of the study, the weak asymptotic method is used to find singular solutions of the shallow water system in both one and two space dimensions. The singular solutions so constructed are allowed to contain Dirac-delta; distributions (Espinosa & Omel'yanov, 2005). The idea is to con- struct complex-valued approximate solutions which become real-valued in the distributional limit. The approach, which extends the range f possible singular solutions, is used to construct solutions which contain combinations of hyperbolic shock waves and Dirac-delta; distributions. It is shown in the second part that the Cauchy problem for Korteweg-de Vries (KdV) type equations is locally ill-posed in a negative Sobolev space. The method is used to construct a solution which does not depend continuously on its initial data in H^{s_epsilon}, s_epsilon = -1/2 - epsilon
Mon, 03 Jun 2013 00:00:00 GMThttp://hdl.handle.net/1956/68022013-06-03T00:00:00ZThe Norwegian Stock Market: - A Local Gaussian Perspectivehttp://hdl.handle.net/1956/6787
The Norwegian Stock Market: - A Local Gaussian Perspective
Lura, Andreas
Master thesis
In this thesis, using daily returns from 18
stocks, oil price, exchange rates and the main
index of the Oslo Stock Exchange over a period of
5 years, we investigate how the Local Gaussian
Correlation can be used
to describe the change in the relationship between
stocks and the market and how it can extend
already established theory
in finance.
Topics covered in this thesis are; risk estimation
by conventional risk measures and a method based
on Local Gaussian Correlation, the Capital Asset
Pricing Model (CAPM), copulas and GARCH as a
description of volatility and as a description of
the marginal distributions for copulas.
Value at Risk and Expected Shortfall are well
established risk measurements in finance.
They are dependent on a good description for the
distribution in the tail, which can be
challenging. These measures only provide one
single number as a description of the risk, this
might be appealing, but does not really provide
detailed information.
By using the theory of CAPM there has been some
attempt to describe the change in risk by using
the so-called conditional moments of the
observations. This approach might be biased, as
the conditional moment
fails to describe the constant correlation and
variances of the Gaussian distribution.
By rather using the local parameters found when
calculating the Local
Gaussian Correlation as a local description of the
beta on our data, there seem to be higher risk in
the upper tail and the lower than in the middle.
However, what differs from the results found by
the previously mentioned approach is that the risk
in the upper tail seems to be higher than in the
lower. This might be explained by very large gains
for the stock market might be followed by a
possible stock market downturn or
even a crash (bubble), while negative values for
the market is less likely to resolve in a sudden
positive boost for the market.
Mon, 03 Jun 2013 00:00:00 GMThttp://hdl.handle.net/1956/67872013-06-03T00:00:00ZDCC-GARCH modeller med ulike avhengighetsstrukturerhttp://hdl.handle.net/1956/6786
DCC-GARCH modeller med ulike avhengighetsstrukturer
Aardal, Helene
Master thesis
Hovedfokuset i denne oppgaven er å finne gode metoder for modellering av volatilitet og avhengighetsstruktur i finansielle porteføljer. En spesifikk multivariat GARCH modell, Dynamic Conditional Correlation (DCC-) GARCH, kombineres med copulaer og par-copula-konstruksjoner for å få en mer fleksibel modell til å modellere nettopp dette.
GARCH-modeller er verktøy for å predikere og analysere volatiliteten i tidsrekker når denne varierer over tid, mens copulaer gir mulighet til å modellere avhengighetsstruktur og marginaler hver for seg.
Flere DCC-GARCH-modeller med ulike avhengighetsstrukturer implementeres i oppgaven; En Copula-DCC-GARCH-modell med multivariat Student t copula, en PCC-DCC-GARCH-modell med Student t-copula for alle par av variable og en PCC-DCC-GARCH-modell der par-copula-konstruksjonen består av både Clayton og Student t copulaer.
Thu, 14 Mar 2013 00:00:00 GMThttp://hdl.handle.net/1956/67862013-03-14T00:00:00ZSemiparametric model selection for copulashttp://hdl.handle.net/1956/6778
Semiparametric model selection for copulas
Jordanger, Lars Arne
Master thesis
This thesis will consider the performance of the cross-validation copula information criterion, xv-CIC, in the realm of finite samples.
The theory leading to the xv-CIC will be outlined, and an analysis will be conducted on an assorted collection of bivariate one-parameter copula models. The restriction to the bivariate case is not a grave one, since more complex d-variate samples can be broken down into a study of conditioned bivariate samples, by the methodology of regular vine-copulas, the pair copula construction and stepwise-semiparametric estimation of parameters.
As a by-product of our analysis, we can give an advice with regard to the selection of model selection method in the semiparametric realm.
Mon, 03 Jun 2013 00:00:00 GMThttp://hdl.handle.net/1956/67782013-06-03T00:00:00ZGenerated Sound Waves in Corrugated Pipeshttp://hdl.handle.net/1956/6693
Generated Sound Waves in Corrugated Pipes
Lillejord, Arve
Master thesis
This paper studies sound waves generated in corrugated pipes by using computational fluid dynam-
ics. The author did not have any prior experience in computational fluid dynamics and therefore
a substantial part was dedicated to theory in computational fluid dynamics. On this background,
this paper aimed to develop the necessary boundary conditions and solver settings in order to
simulate the problem using OpenFOAM. The main result consists of two cases, where the first
case studied the flow over a single corrugation and the second case a study of the flow through a
corrugated pipe. It was found that the first case was able to predict frequencies matching previous
work. The second case was not able to produce the features connected to singing in corrugated
pipes. This is largely attributed to how the inlet boundary condition was set.
Fri, 01 Jun 2012 00:00:00 GMThttp://hdl.handle.net/1956/66932012-06-01T00:00:00ZMisoppfatningar i algebra på ungdomsskulen - ei diagnostisk tilnærminghttp://hdl.handle.net/1956/6692
Misoppfatningar i algebra på ungdomsskulen - ei diagnostisk tilnærming
Hompland, Anne Berit
Master thesis
Denne oppgåva handlar om elevar i 10.klasse sitt arbeid med algebra. Meir spesifikt har eg
tatt utgangspunkt i ei kvalitativ tilnærming for å undersøka misoppfatningane dei har innan
emnet. Vidare har eg sett på korleis ein med dette utgangspunktet kan handsama misoppfatningane i undervisinga ved hjelp av diagnostisk arbeidsmetode. Med desse måla for
arbeidet valde eg å gjera eigne undersøkingar for å skaffa materiale.
Konstruktivistiske læringsteoriar utgjer det teoretiske rammeverket for oppgåva. Meir
spesifikt omhandlar det omgrep som misoppfatningar, diagnostiske oppgåver og diagnostisk undervising. Med dette som bakgrunn har eg tatt utgangspunkt i tidlegare forsking om
misoppfatningar innan algebra og nytta det i mine eigne undersøkingar.
Sjølv om det kvalitative aspektet var motivasjonen for å gå i gang med undersøkingane, valde eg å nytta metodetriangulering fordi det gav ei meir heilskapleg tilnærming til det eg ville undersøka. Dei kvalitative delane var intervju med tre elevar og diagnostisk undervising i ei gruppe på 14 elevar. Før dette hadde eg hatt ei kvantitativ kartleggingsprøve på heile 10.trinn. I etterkant gjennomførte eg ein ny kartleggingsprøve på gruppa med 14 elevar.
Resultata, både frå kartleggingsprøvane og intervjua, synte ei elevgruppe med store
utfordringar innan algebra, noko som gjenspeglar det både PISA- og TIMSS-undersøkingar frå tidlegare år alt har peika på. Tidlegare dokumenterte misoppfatningar, både frå norske og
utanlandske undersøkingar, synte seg hjå desse elevane. I arbeidet med diagnostisk
undervising retta eg fokuset direkte inn mot misoppfatningane for om mogeleg å bearbeida
dei. Kartleggingsprøva i etterkant synte eit svakt resultat med tanke på forbetringar. Men som
drøftinga i samband med dette syner, kan det ha andre orsakar enn akkurat mi tilnærming til
diagnostisk arbeid med algebra.
Avslutningsvis peiker eg på tre punkt som eg ser på som hovudutfordringar i arbeidet med
misoppfatningar i algebra; det aritmetiske grunnlaget, forståing av bokstavsymbola og elevane
sin bruk av intuitive strategiar. Sjølv om utvalet mitt ikkje er stort nok til å seia om dette er generelle tendensar, meiner eg å ha grunnlag for å seia at dette er utfordringar som er viktige for matematikklærarar å kjenna til.
Thu, 24 May 2012 00:00:00 GMThttp://hdl.handle.net/1956/66922012-05-24T00:00:00ZNivådeling i matematikk på ungdomsskolen - en kvalitativ undersøkelsehttp://hdl.handle.net/1956/6691
Nivådeling i matematikk på ungdomsskolen - en kvalitativ undersøkelse
Tindeland, Anita
Master thesis
I Kunnskapløftet, K06, legges det stor vekt på tilpasset opplæring til hver enkelt elev. Flere ungdomsskoler har innført nivådeling i matematikk som et tiltak for å imøtekomme dette kravet. I Norge står enhetsskoletanken sterkt, og nivådeling er et svært omstridt tema. Både forskere og politikere har delte meninger på området. Jeg har, ved hjelp av det kvalitative forskningsintervjuet, sett nærmere på hvordan nivådeling gjennomføres på to skoler i bergensområdet. Problemstillingen jeg har holdt meg til er: Hvilke konsekvenser har det på det sosiale, personlige og organisatoriske plan, å nivådele matematikkundervisningen på ungdomstrinnet? I arbeidet med oppgaven har jeg gjort svært mange og komplekse funn. Jeg vil derfor ikke komme med et endelig svar for eller mot nivådeling. I teksten belyser jeg hvilke positive og negative konsekvenser nivådeling har for elever og læreres skolehverdag. Her har jeg blant annet sett nærmere på elever og læreres oppfatning av ordningens innvirkning på mobbing, sosial status, motivasjon, mestring og forsvarstrategier, organisering av gruppeinndeling og undervisning. Jeg finner at det er et forbedringspotensiale ved de ordningene jeg har undersøkt. Spesielt gjelder dette hvordan undervisningen legges opp på de ulike gruppene, og hvordan mulighetene for tilpasning og variasjon utnyttes. Vellykket nivådeling - om så finnes - avhenger av en rekke faktorer; elevmassen, lærerne, tilgjengelige ressurser og hvilke undervisningsmetoder som brukes, for å nevne noe. I denne studien har jeg belyst noen av konsekvensene av en slik ordning ved to spesifikke skoler. Jeg finner at det er både positive og negative sider ved de undersøkte ordningene, og jeg presenterer avslutningsvis noen forbedringsforslag som muligens kan bidra til å gjøre dem enda bedre.
Wed, 30 May 2012 00:00:00 GMThttp://hdl.handle.net/1956/66912012-05-30T00:00:00ZAdaptive parameterization of electric conductivity in inversion of electromagnetic datahttp://hdl.handle.net/1956/6687
Adaptive parameterization of electric conductivity in inversion of electromagnetic data
Ek, Torbjørn Helland
Master thesis
We describe a methodology developed for 3-D parameter identification, with focus on large-scale applications such as monitoring subsea oil production and geothermal systems. The methodology is designed to handle challenges related to low parameter sensitivity, nonuniqueness of the inverse solutions and costly numerical calculations. A reduced, composite parameter representation is chosen to meet these challenges. Our contributions to the methodology involves choosing a reduced representation with radial basis functions, to maintain a low number of parameters. We also propose the use of a 1. order selection measure in the refinement process to reduce the computational costs. The performance of the proposed changes in the methodology is illustrated in a series of examples for estimating the change in electric conductivity from time-lapse electromagnetic observations. The results show some limitetions regarding the accuracy of the 1. order selection measure. For the investigated numerical examples, radial basis functions, together with the described methodology, effectively provide an estimated of the electric conductivity field using electromagnetic measurements.
Fri, 01 Jun 2012 00:00:00 GMThttp://hdl.handle.net/1956/66872012-06-01T00:00:00ZGonality of Points in Brill-Noether Loci of the Moduli Space of Curveshttp://hdl.handle.net/1956/6686
Gonality of Points in Brill-Noether Loci of the Moduli Space of Curves
Sæbø, Gard
Master thesis
In this masterthesis we study the geometry of the
points in the Brill-Noether locus. Typically, we
want to study the gonality of points in this
locus, that can be seen as projective curves. We
also study what curves on K3s, which corresponds
to smooth curves in the Hilbert scheme.; I denne masteroppgaven studerer vi geometrien til
punktene i Brill-Noether lokuset. Et typisk
problem er å finne gonaliteten til disse punktene,
som kan bli sett på som projektive kurver. Vi skal
også studere hvilke kurver på K3 flater som
korresponderer til glatte punkter i
Hilbertskjemaet.
Thu, 31 May 2012 00:00:00 GMThttp://hdl.handle.net/1956/66862012-05-31T00:00:00ZImpact of compressibility in vertically integrated models for CO2 storagehttp://hdl.handle.net/1956/6685
Impact of compressibility in vertically integrated models for CO2 storage
Reistad, Silje Rognsvåg
Master thesis
Recent work have shown that simplified models obtained by vertical integration give reasonable approximations for CO2 migration. Most of the work previously done on this topic assumes that CO2 is an incompressible fluid, or they work with a quasi compressibility, where they consider horizontal compressibility but not vertical. However, CO2 is highly compressible and large variations in CO2 density can be expected due to pressure increases near the injection well. In addition, long term CO2 migration to shallower depths can result in significantly lower density as dictated by geothermal gradients and regional pressure conditions. In this study, we outline the mathematical and numerical approach to solve the coupled vertically integrated models with density variation. Mathematical models for CO2 storage are considered in which the equations for mass conservation and Darcy's law are vertically integrated and coupled with an equation of state for CO2 density. Based on these, a pressure equation is derived. This pressure equation is then discretised with the use of an implicit control volume method and a partially implicit Lax-Friedrichs' method. A solution approach based on an IMPES approach is suggested.
We have unfortunately not been able to solve this problem, as there has been some difficulties with the boundary values in the code, but we suggest a plan for further work for a working code.
Fri, 01 Jun 2012 00:00:00 GMThttp://hdl.handle.net/1956/66852012-06-01T00:00:00ZCritical Measures and Parameter Space of Jenkins-Strebel Quadratic Differentialshttp://hdl.handle.net/1956/6684
Critical Measures and Parameter Space of Jenkins-Strebel Quadratic Differentials
Frolova, Anastasia
Master thesis
The thesis is devoted to applications of the
theory of quadratic differentials
to the problems of construction
of critical measures, that provide critical values
of the logarithmic energy of a charge system on
the complex plane. We observe the properties of
supports of such measures. We consider the problem
of describing parameter space of Jenkins-Strebel
differentials, which is related to the problem of
describing critical measures.
Mon, 07 May 2012 00:00:00 GMThttp://hdl.handle.net/1956/66842012-05-07T00:00:00ZVascular responses to radiotherapy and androgen-deprivation therapy in experimental prostate cancerhttp://hdl.handle.net/1956/6664
Vascular responses to radiotherapy and androgen-deprivation therapy in experimental prostate cancer
Røe, Kathrine; Mikalsen, Lars T. G.; Kogel, Albert J. van der; Bussink, Johan; Lyng, Heidi; Olsen, Dag R.
Peer reviewed; Journal article
<p>Background: Radiotherapy (RT) and androgen-deprivation therapy (ADT) are standard treatments for advanced prostate cancer (PC). Tumor vascularization is recognized as an important physiological feature likely to impact on both RT and ADT response, and this study therefore aimed to characterize the vascular responses to RT and ADT in experimental PC.</p> <p>Methods: Using mice implanted with CWR22 PC xenografts, vascular responses to RT and ADT by castration were visualized in vivo by DCE MRI, before contrast-enhancement curves were analyzed both semi-quantitatively and by pharmacokinetic modeling. Extracted image parameters were correlated to the results from ex vivo quantitative fluorescent immunohistochemical analysis (qIHC) of tumor vascularization (9 F1), perfusion (Hoechst 33342), and hypoxia (pimonidazole), performed on tissue sections made from tumors excised directly after DCE MRI.</p> <p>Results: Compared to untreated (Ctrl) tumors, an improved and highly functional vascularization was detected in androgen-deprived (AD) tumors, reflected by increases in DCE MRI parameters and by increased number of vessels (VN), vessel density ( VD), and vessel area fraction ( VF) from qIHC. Although total hypoxic fractions ( HF) did not change, estimated acute hypoxia scores ( AHS) – the proportion of hypoxia staining within 50 μm from perfusion staining – were increased in AD tumors compared to in Ctrl tumors. Five to six months after ADT renewed castration-resistant (CR) tumor growth appeared with an even further enhanced tumor vascularization. Compared to the large vascular changes induced by ADT, RT induced minor vascular changes. Correlating DCE MRI and qIHC parameters unveiled the semi-quantitative parameters area under curve ( AUC) from initial time-points to strongly correlate with VD and VF, whereas estimation of vessel size ( VS) by DCE MRI required pharmacokinetic modeling. HF was not correlated to any DCE MRI parameter, however, AHS may be estimated after pharmacokinetic modeling. Interestingly, such modeling also detected tumor necrosis very strongly.</p> <p>Conclusions: DCE MRI reliably allows non-invasive assessment of tumors’ vascular function. The findings of increased tumor vascularization after ADT encourage further studies into whether these changes are beneficial for combined RT, or if treatment with anti-angiogenic therapy may be a strategy to improve the therapeutic efficacy of ADT in advanced PC.</p>
Wed, 23 May 2012 00:00:00 GMThttp://hdl.handle.net/1956/66642012-05-23T00:00:00ZMultiscale simulation of flow and heat transport in fractured geothermal reservoirshttp://hdl.handle.net/1956/6593
Multiscale simulation of flow and heat transport in fractured geothermal reservoirs
Sandve, Tor Harald
Doctoral thesis
Mon, 04 Mar 2013 00:00:00 GMThttp://hdl.handle.net/1956/65932013-03-04T00:00:00ZLocal Likelihoodhttp://hdl.handle.net/1956/6583
Local Likelihood
Otneim, Håkon
Master thesis
Methods for probability density estimation are
traditionally classified as either parametric or
non-parametric. Fitting a parametric model to
observations is generally a good idea when we have
sufficient information on the origin of our data;
if not, we must turn to non-parametric methods,
usually at the cost of poorer performance.
This thesis discusses local maximum likelihood
estimation of probability density functions, which
can be regarded as a compromise between the two
mindsets. The idea is to fit a parametric model
locally, that is, to let the parameters and their
estimates depend on the location. If the chosen
model is close to the true, unknown density, we
keep much of the appealing properties of a full
parametric approach. On the other hand, local
likelihood density estimates have performance
comparable to well known non-parametric methods,
even though the locally fitted parametric model
differs from the true density in a global sense.
Although traditional methods withstand the test of
time as excellent options in many situations, the
local maximum likelihood estimator opens up a
range of applications. Hjort and Jones [1996], who
will serve as the main reference for this thesis,
call it semi-parametric density estimation, as it
is particularly useful when we have partial
knowledge on the shape of the unknown density, but
not enough to trust the ordinary, global maximum
likelihood estimates. Further, many have built on
the idea of locally parametric estimation to
applications beyond just density estimation, some
of whom have been mentioned and included as
references throughout the thesis.
One-dimensional density estimation will, however,
be the primary focus here, with particular
emphasis on large sample theory. The main results
concern asymptotic bias, which is shown to have a
larger order than the bias of traditional kernel
estimation as the sample size increases to
infinity, and the bandwidth decreases towards
zero. Nonetheless, in practical situations with
reasonable sample sizes, the local likelihood
estimator is shown to perform very well, with an
appealing robustness against under- and
oversmoothing. Indeed, no experiment performed
show signs of deterioration of local likelihood
estimates compared to kernel estimation as the
sample size grows.
Mon, 23 Apr 2012 00:00:00 GMThttp://hdl.handle.net/1956/65832012-04-23T00:00:00ZGAMLSS-modeller i bilforsikringhttp://hdl.handle.net/1956/6582
GAMLSS-modeller i bilforsikring
Røyrane-Løtvedt, Hallvard
Master thesis
I denne oppgaven tester jeg ulike modeller for prediksjon av total skadeutbetaling fra forsikringsselskap til forsikringstaker i et poliseår. Modellene som testes hører til rammeverket Generalized Additive Models for Location, Shape and Scale - GAMLSS - introdusert av Rigby og Stasinopoulos (2001). Data brukt i oppgaven er hentet fra et norsk forsikringsselskap, og består av informasjon om poliser og skader i bilforsikring i årene 2000-2005. Ved hjelp av kun 3 forklaringsvariabler; årstall, bilalder og personalder, viser jeg i denne oppgaven at valg av statistisk modell er avgjørende for prediksjonene av skadeutbetalingen (kapittel 9). Videre tester jeg ut hvordan modellprediksjonene kan brukes til å lage en realistisk prismodell, og hvordan prismodellen gir ulike resultater for de ulike prediksjonsmodellene (kapittel 10).
Total skadeutbetaling deles naturlig inn i skadefrekvens og skadepris. Jeg tester i oppgaven både modeller som modellerer disse separat, og modeller som modellerer total skadeutbetaling direkte. Jeg vil argumentere for at de direkte modellene er å foretrekke. Modellen som anbefales er en Zero-Adjusted Inverse Gaussian - ZAIG-modell, der forklaringsvariablenes funksjonelle form er valgt slik at AIC blir så lav som mulig. En ZAIG-fordelt stokastisk variabel tar verdien 0 med sannsynlighet psi og følger en Invers-Gaussisk-fordeling med sannsynlighet (1-psi). Skadepriser er såpass skjevt fordelt at det må en ekstremt skjev sannsynlighetsfordeling, som den Invers-Gaussiske, til, for å beskrive dem. Jeg vil også i oppgaven argumentere for at valg av sannsynlighetsfordeling har stor betydning for kvaliteten på prediksjonene.
Fri, 01 Jun 2012 00:00:00 GMThttp://hdl.handle.net/1956/65822012-06-01T00:00:00ZStatistiske modeller basert på skjulte Markovkjeder i kontinuerlig tidhttp://hdl.handle.net/1956/6581
Statistiske modeller basert på skjulte Markovkjeder i kontinuerlig tid
Harjo, Lisbet Lien
Master thesis
I denne oppgaven har jeg tatt for meg situasjoner der observasjoner over tid kan tenkes å være generert ved en Markovkjede,
og studert innvirkningen av ulike kovariater på intensitetene, med både binære og tidavhengige kovariater.
Jeg har også sett på situasjoner der tilstandene i Markovkjeden kan være observert feil, slik at man ikke virkelig observerer
de underliggende tilstandene, men bare feilklassifiserte tilstander bestemt av en sannsynlighetsfordeling.
Til slutt har jeg også testet ut en aktuell pakke for programvaren \rkode{R} som er blitt utarbeidet for å kunne regne på denne typen problemstillinger.
Fri, 01 Jun 2012 00:00:00 GMThttp://hdl.handle.net/1956/65812012-06-01T00:00:00ZCopulas and Local Gaussian Correlationhttp://hdl.handle.net/1956/6580
Copulas and Local Gaussian Correlation
Nordbø, Tommy Neverdahl
Master thesis
We are looking at copula models and dependence measures. Especially the recently developed local dependence measure called local Gaussian correlation (LGC) is presented, and its connection with the copula theory is explored. Theoretical LGC values of some famous (and not so famous) copula models are derived, and used to make plots that shows the dependence structure of the copula. At the end we are outlining some ways of using this dependence measure for copula selection and goodness-of-fit tests.
Wed, 30 May 2012 00:00:00 GMThttp://hdl.handle.net/1956/65802012-05-30T00:00:00ZPasser og linjal, origami og Galoisteorihttp://hdl.handle.net/1956/6579
Passer og linjal, origami og Galoisteori
Vasdal, Thomas Arneberg
Master thesis
Denne oppgaven prøver å belyse konstruksjoner med passer og linjal og origami ved hjelp av Galois- teori. De tre klassiske problemene: Kubens fordobling, sirkelens kvadratur og vinkelens tredeling er sentrale. Oppgaven er myntet særlig på lærere med lektor kompetanse i matematikk. Det er en fordel å tatt introduksjonskurset i abstrakt algebra.
Wed, 01 Jun 2011 00:00:00 GMThttp://hdl.handle.net/1956/65792011-06-01T00:00:00ZTestmetoder for identifisering av publikasjonsbias i metaanalyserhttp://hdl.handle.net/1956/6578
Testmetoder for identifisering av publikasjonsbias i metaanalyser
Gjerdevik, Miriam
Master thesis
Testmetoder for identifisering av publikasjonsbias i metaanalyser.
Wed, 24 Oct 2012 00:00:00 GMThttp://hdl.handle.net/1956/65782012-10-24T00:00:00Z